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2026-01-01
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<p>Last updated on<strong>September 26, 2025</strong></p>
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<p>Last updated on<strong>September 26, 2025</strong></p>
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<p>We use the derivative of ln(sin x), which is cot(x), as a tool for understanding how the logarithmic function changes in response to a slight change in x. Derivatives help us calculate profit or loss in real-life situations. We will now discuss the derivative of ln(sin x) in detail.</p>
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<p>We use the derivative of ln(sin x), which is cot(x), as a tool for understanding how the logarithmic function changes in response to a slight change in x. Derivatives help us calculate profit or loss in real-life situations. We will now discuss the derivative of ln(sin x) in detail.</p>
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<h2>What is the Derivative of ln(sin x)?</h2>
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<h2>What is the Derivative of ln(sin x)?</h2>
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<p>We now understand the derivative<a>of</a>ln(sin x). It is commonly represented as d/dx (ln(sin x)) or (ln(sin x))', and its value is cot x. The<a>function</a>ln(sin x) has a clearly defined derivative, indicating it is differentiable within its domain.</p>
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<p>We now understand the derivative<a>of</a>ln(sin x). It is commonly represented as d/dx (ln(sin x)) or (ln(sin x))', and its value is cot x. The<a>function</a>ln(sin x) has a clearly defined derivative, indicating it is differentiable within its domain.</p>
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<p>The key concepts are mentioned below:</p>
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<p>The key concepts are mentioned below:</p>
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<p>Logarithmic Function: ln(x) is the natural logarithm function.</p>
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<p>Logarithmic Function: ln(x) is the natural logarithm function.</p>
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<p>Chain Rule: Rule for differentiating ln(sin x) (since it involves a composition of functions).</p>
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<p>Chain Rule: Rule for differentiating ln(sin x) (since it involves a composition of functions).</p>
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<p>Cotangent Function: cot(x) = 1/tan(x).</p>
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<p>Cotangent Function: cot(x) = 1/tan(x).</p>
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<h2>Derivative of ln(sin x) Formula</h2>
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<h2>Derivative of ln(sin x) Formula</h2>
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<p>The derivative of ln(sin x) can be denoted as d/dx (ln(sin x)) or (ln(sin x))'. The<a>formula</a>we use to differentiate ln(sin x) is: d/dx (ln(sin x)) = cot x (or) (ln(sin x))' = cot x</p>
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<p>The derivative of ln(sin x) can be denoted as d/dx (ln(sin x)) or (ln(sin x))'. The<a>formula</a>we use to differentiate ln(sin x) is: d/dx (ln(sin x)) = cot x (or) (ln(sin x))' = cot x</p>
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<p>The formula applies to all x where sin(x) > 0</p>
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<p>The formula applies to all x where sin(x) > 0</p>
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<h2>Proofs of the Derivative of ln(sin x)</h2>
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<h2>Proofs of the Derivative of ln(sin x)</h2>
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<p>We can derive the derivative of ln(sin x) using proofs. To show this, we will use the trigonometric identities along with the rules of differentiation. There are several methods we use to prove this, such as:</p>
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<p>We can derive the derivative of ln(sin x) using proofs. To show this, we will use the trigonometric identities along with the rules of differentiation. There are several methods we use to prove this, such as:</p>
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<ul><li>By First Principle </li>
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<ul><li>By First Principle </li>
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<li>Using Chain Rule </li>
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<li>Using Chain Rule </li>
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<li>Using Product Rule</li>
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<li>Using Product Rule</li>
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</ul><p>We will now demonstrate that the differentiation of ln(sin x) results in cot x using the above-mentioned methods:</p>
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</ul><p>We will now demonstrate that the differentiation of ln(sin x) results in cot x using the above-mentioned methods:</p>
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<h2><strong>By First Principle</strong></h2>
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<h2><strong>By First Principle</strong></h2>
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<p>The derivative of ln(sin x) can be proved using the First Principle, which expresses the derivative as the limit of the difference<a>quotient</a>. To find the derivative of ln(sin x) using the first principle, we will consider f(x) = ln(sin x). Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1) Given that f(x) = ln(sin x), we write f(x + h) = ln(sin(x + h)). Substituting these into<a>equation</a>(1), f'(x) = limₕ→₀ [ln(sin(x + h)) - ln(sin x)] / h = limₕ→₀ ln([sin(x + h) / sin x]) / h Using the logarithmic identity ln(a) - ln(b) = ln(a/b), f'(x) = limₕ→₀ [ln(1 + (sin(x + h) - sin x)/sin x)] / h Using the limit definition, f'(x) = limₕ→₀ [(sin(x + h) - sin x)/(h sin x)] Using the identity sin(A + h) - sin A = 2 cos((2A + h)/2) sin(h/2), f'(x) = limₕ→₀ [cos(x + h/2) sin(h/2)/sin x] / (h/2) Using limit formulas, limₕ→₀ sin(h/2)/(h/2) = 1. f'(x) = cos x/sin x = cot x.</p>
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<p>The derivative of ln(sin x) can be proved using the First Principle, which expresses the derivative as the limit of the difference<a>quotient</a>. To find the derivative of ln(sin x) using the first principle, we will consider f(x) = ln(sin x). Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1) Given that f(x) = ln(sin x), we write f(x + h) = ln(sin(x + h)). Substituting these into<a>equation</a>(1), f'(x) = limₕ→₀ [ln(sin(x + h)) - ln(sin x)] / h = limₕ→₀ ln([sin(x + h) / sin x]) / h Using the logarithmic identity ln(a) - ln(b) = ln(a/b), f'(x) = limₕ→₀ [ln(1 + (sin(x + h) - sin x)/sin x)] / h Using the limit definition, f'(x) = limₕ→₀ [(sin(x + h) - sin x)/(h sin x)] Using the identity sin(A + h) - sin A = 2 cos((2A + h)/2) sin(h/2), f'(x) = limₕ→₀ [cos(x + h/2) sin(h/2)/sin x] / (h/2) Using limit formulas, limₕ→₀ sin(h/2)/(h/2) = 1. f'(x) = cos x/sin x = cot x.</p>
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<h2><strong>Using Chain Rule</strong></h2>
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<h2><strong>Using Chain Rule</strong></h2>
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<p>To prove the differentiation of ln(sin x) using the chain rule, We use the formula: Let u = sin x, then ln(u) Differentiate using the chain rule: d(ln(u))/dx = 1/u * du/dx So, d(ln(sin x))/dx = 1/sin x * cos x = cot x.</p>
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<p>To prove the differentiation of ln(sin x) using the chain rule, We use the formula: Let u = sin x, then ln(u) Differentiate using the chain rule: d(ln(u))/dx = 1/u * du/dx So, d(ln(sin x))/dx = 1/sin x * cos x = cot x.</p>
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<h2><strong>Using Product Rule</strong></h2>
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<h2><strong>Using Product Rule</strong></h2>
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<p>We will now prove the derivative of ln(sin x) using the<a>product</a>rule. Let y = ln(sin x) = ln u, where u = sin x. Differentiate using the chain rule: dy/dx = 1/u * du/dx = 1/sin x * cos x = cot x.</p>
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<p>We will now prove the derivative of ln(sin x) using the<a>product</a>rule. Let y = ln(sin x) = ln u, where u = sin x. Differentiate using the chain rule: dy/dx = 1/u * du/dx = 1/sin x * cos x = cot x.</p>
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<h2>Higher-Order Derivatives of ln(sin x)</h2>
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<h2>Higher-Order Derivatives of ln(sin x)</h2>
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<p>When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be a little tricky.</p>
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<p>When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be a little tricky.</p>
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<p>To understand them better, think of a car where the speed changes (first derivative) and the<a>rate</a>at which the speed changes (second derivative) also changes. Higher-order derivatives make it easier to understand functions like ln(sin x).</p>
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<p>To understand them better, think of a car where the speed changes (first derivative) and the<a>rate</a>at which the speed changes (second derivative) also changes. Higher-order derivatives make it easier to understand functions like ln(sin x).</p>
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<p>For the first derivative of a function, we write f′(x), which indicates how the function changes or its slope at a certain point. The second derivative is derived from the first derivative, which is denoted using f′′ (x) Similarly, the third derivative, f′′′(x) is the result of the second derivative and this pattern continues.</p>
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<p>For the first derivative of a function, we write f′(x), which indicates how the function changes or its slope at a certain point. The second derivative is derived from the first derivative, which is denoted using f′′ (x) Similarly, the third derivative, f′′′(x) is the result of the second derivative and this pattern continues.</p>
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<p>For the nth Derivative of ln(sin x), we generally use fⁿ(x) for the nth derivative of a function f(x) which tells us the change in the rate of change. (continuing for higher-order derivatives).</p>
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<p>For the nth Derivative of ln(sin x), we generally use fⁿ(x) for the nth derivative of a function f(x) which tells us the change in the rate of change. (continuing for higher-order derivatives).</p>
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<h2>Special Cases:</h2>
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<h2>Special Cases:</h2>
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<p>When x is 0, the derivative is undefined because ln(sin x) is undefined for sin x = 0.</p>
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<p>When x is 0, the derivative is undefined because ln(sin x) is undefined for sin x = 0.</p>
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<p>When x is π/2, the derivative of ln(sin x) = cot(π/2), which is 0.</p>
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<p>When x is π/2, the derivative of ln(sin x) = cot(π/2), which is 0.</p>
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<h2>Common Mistakes and How to Avoid Them in Derivatives of ln(sin x)</h2>
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<h2>Common Mistakes and How to Avoid Them in Derivatives of ln(sin x)</h2>
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<p>Students frequently make mistakes when differentiating ln(sin x). These mistakes can be resolved by understanding the proper solutions. Here are a few common mistakes and ways to solve them:</p>
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<p>Students frequently make mistakes when differentiating ln(sin x). These mistakes can be resolved by understanding the proper solutions. Here are a few common mistakes and ways to solve them:</p>
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<h3>Problem 1</h3>
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<h3>Problem 1</h3>
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<p>Calculate the derivative of ln(sin x)·cos x</p>
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<p>Calculate the derivative of ln(sin x)·cos x</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>Here, we have f(x) = ln(sin x)·cos x. Using the product rule, f'(x) = u′v + uv′ In the given equation, u = ln(sin x) and v = cos x. Let’s differentiate each term, u′ = d/dx (ln(sin x)) = cot x v′ = d/dx (cos x) = -sin x Substituting into the given equation, f'(x) = (cot x)·(cos x) + (ln(sin x))·(-sin x) Let’s simplify terms to get the final answer, f'(x) = cot x cos x - ln(sin x) sin x Thus, the derivative of the specified function is cot x cos x - ln(sin x) sin x.</p>
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<p>Here, we have f(x) = ln(sin x)·cos x. Using the product rule, f'(x) = u′v + uv′ In the given equation, u = ln(sin x) and v = cos x. Let’s differentiate each term, u′ = d/dx (ln(sin x)) = cot x v′ = d/dx (cos x) = -sin x Substituting into the given equation, f'(x) = (cot x)·(cos x) + (ln(sin x))·(-sin x) Let’s simplify terms to get the final answer, f'(x) = cot x cos x - ln(sin x) sin x Thus, the derivative of the specified function is cot x cos x - ln(sin x) sin x.</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>We find the derivative of the given function by dividing the function into two parts.</p>
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<p>We find the derivative of the given function by dividing the function into two parts.</p>
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<p>The first step is finding its derivative and then combining them using the product rule to get the final result.</p>
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<p>The first step is finding its derivative and then combining them using the product rule to get the final result.</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h3>Problem 2</h3>
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<h3>Problem 2</h3>
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<p>A certain wave pattern in a physics experiment is represented by the function y = ln(sin x) where y represents the intensity of the wave at a point x. If x = π/3, measure the rate of change of intensity.</p>
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<p>A certain wave pattern in a physics experiment is represented by the function y = ln(sin x) where y represents the intensity of the wave at a point x. If x = π/3, measure the rate of change of intensity.</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>We have y = ln(sin x) (wave intensity)...(1) Now, we will differentiate the equation (1) Take the derivative ln(sin x): dy/dx = cot x Given x = π/3 (substitute this into the derivative) cot(π/3) = 1/tan(π/3) cot(π/3) = 1/(√3) Hence, we get the rate of change of intensity at x = π/3 as 1/√3.</p>
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<p>We have y = ln(sin x) (wave intensity)...(1) Now, we will differentiate the equation (1) Take the derivative ln(sin x): dy/dx = cot x Given x = π/3 (substitute this into the derivative) cot(π/3) = 1/tan(π/3) cot(π/3) = 1/(√3) Hence, we get the rate of change of intensity at x = π/3 as 1/√3.</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>We find the rate of change of intensity at x = π/3 as 1/√3, which means that at a given point, the intensity of the wave would change at this rate.</p>
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<p>We find the rate of change of intensity at x = π/3 as 1/√3, which means that at a given point, the intensity of the wave would change at this rate.</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h3>Problem 3</h3>
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<h3>Problem 3</h3>
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<p>Derive the second derivative of the function y = ln(sin x).</p>
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<p>Derive the second derivative of the function y = ln(sin x).</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>The first step is to find the first derivative, dy/dx = cot x...(1) Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [cot x] Here we use the derivative of cot x, d²y/dx² = -csc²x Therefore, the second derivative of the function y = ln(sin x) is -csc²x.</p>
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<p>The first step is to find the first derivative, dy/dx = cot x...(1) Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [cot x] Here we use the derivative of cot x, d²y/dx² = -csc²x Therefore, the second derivative of the function y = ln(sin x) is -csc²x.</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>We use the step-by-step process, where we start with the first derivative.</p>
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<p>We use the step-by-step process, where we start with the first derivative.</p>
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<p>We then differentiate cot x to find the second derivative.</p>
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<p>We then differentiate cot x to find the second derivative.</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h3>Problem 4</h3>
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<h3>Problem 4</h3>
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<p>Prove: d/dx (ln(sin²x)) = 2 cot x.</p>
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<p>Prove: d/dx (ln(sin²x)) = 2 cot x.</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>Let’s start using the chain rule: Consider y = ln(sin²x) = 2 ln(sin x) To differentiate, we use the chain rule: dy/dx = 2 d/dx (ln(sin x)) Since the derivative of ln(sin x) is cot x, dy/dx = 2 cot x Hence proved.</p>
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<p>Let’s start using the chain rule: Consider y = ln(sin²x) = 2 ln(sin x) To differentiate, we use the chain rule: dy/dx = 2 d/dx (ln(sin x)) Since the derivative of ln(sin x) is cot x, dy/dx = 2 cot x Hence proved.</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>In this step-by-step process, we used the chain rule to differentiate the equation.</p>
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<p>In this step-by-step process, we used the chain rule to differentiate the equation.</p>
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<p>Then, we replace the derivative of ln(sin x) with cot x.</p>
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<p>Then, we replace the derivative of ln(sin x) with cot x.</p>
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<p>As a final step, we simplify to derive the equation.</p>
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<p>As a final step, we simplify to derive the equation.</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h3>Problem 5</h3>
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<h3>Problem 5</h3>
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<p>Solve: d/dx (ln(sin x)/x)</p>
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<p>Solve: d/dx (ln(sin x)/x)</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>To differentiate the function, we use the quotient rule: d/dx (ln(sin x)/x) = (d/dx (ln(sin x))·x - ln(sin x)·d/dx(x))/x² We will substitute d/dx (ln(sin x)) = cot x and d/dx (x) = 1 = (cot x·x - ln(sin x)·1)/x² = (x cot x - ln(sin x))/x² Therefore, d/dx (ln(sin x)/x) = (x cot x - ln(sin x))/x²</p>
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<p>To differentiate the function, we use the quotient rule: d/dx (ln(sin x)/x) = (d/dx (ln(sin x))·x - ln(sin x)·d/dx(x))/x² We will substitute d/dx (ln(sin x)) = cot x and d/dx (x) = 1 = (cot x·x - ln(sin x)·1)/x² = (x cot x - ln(sin x))/x² Therefore, d/dx (ln(sin x)/x) = (x cot x - ln(sin x))/x²</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>In this process, we differentiate the given function using the quotient rule.</p>
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<p>In this process, we differentiate the given function using the quotient rule.</p>
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<p>As a final step, we simplify the equation to obtain the final result.</p>
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<p>As a final step, we simplify the equation to obtain the final result.</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h2>FAQs on the Derivative of ln(sin x)</h2>
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<h2>FAQs on the Derivative of ln(sin x)</h2>
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<h3>1.Find the derivative of ln(sin x).</h3>
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<h3>1.Find the derivative of ln(sin x).</h3>
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<p>Using the chain rule for ln(sin x) gives: d/dx (ln(sin x)) = cot x (simplified)</p>
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<p>Using the chain rule for ln(sin x) gives: d/dx (ln(sin x)) = cot x (simplified)</p>
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<h3>2.Can we use the derivative of ln(sin x) in real life?</h3>
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<h3>2.Can we use the derivative of ln(sin x) in real life?</h3>
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<p>Yes, we can use the derivative of ln(sin x) in real life in calculating the rate of change of any motion, especially in fields such as mathematics, physics, and engineering.</p>
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<p>Yes, we can use the derivative of ln(sin x) in real life in calculating the rate of change of any motion, especially in fields such as mathematics, physics, and engineering.</p>
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<h3>3.Is it possible to take the derivative of ln(sin x) at the point where x = 0?</h3>
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<h3>3.Is it possible to take the derivative of ln(sin x) at the point where x = 0?</h3>
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<p>No, x = 0 is a point where ln(sin x) is undefined, so it is impossible to take the derivative at these points (since the function does not exist there).</p>
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<p>No, x = 0 is a point where ln(sin x) is undefined, so it is impossible to take the derivative at these points (since the function does not exist there).</p>
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<h3>4.What rule is used to differentiate ln(sin x)/x?</h3>
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<h3>4.What rule is used to differentiate ln(sin x)/x?</h3>
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<p>We use the quotient rule to differentiate ln(sin x)/x, d/dx (ln(sin x)/x) = (x cot x - ln(sin x))/x².</p>
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<p>We use the quotient rule to differentiate ln(sin x)/x, d/dx (ln(sin x)/x) = (x cot x - ln(sin x))/x².</p>
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<h3>5.Are the derivatives of ln(sin x) and ln(cos x) the same?</h3>
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<h3>5.Are the derivatives of ln(sin x) and ln(cos x) the same?</h3>
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<p>No, they are different. The derivative of ln(sin x) is cot x, while the derivative of ln(cos x) is -tan x.</p>
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<p>No, they are different. The derivative of ln(sin x) is cot x, while the derivative of ln(cos x) is -tan x.</p>
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<h2>Important Glossaries for the Derivative of ln(sin x)</h2>
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<h2>Important Glossaries for the Derivative of ln(sin x)</h2>
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<ul><li><strong>Derivative:</strong>The derivative of a function indicates how the given function changes in response to a slight change in x.</li>
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<ul><li><strong>Derivative:</strong>The derivative of a function indicates how the given function changes in response to a slight change in x.</li>
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</ul><ul><li><strong>Logarithmic Function:</strong>A function that uses logarithms, often denoted by ln(x).</li>
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</ul><ul><li><strong>Logarithmic Function:</strong>A function that uses logarithms, often denoted by ln(x).</li>
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</ul><ul><li><strong>Chain Rule:</strong>A rule for differentiating compositions of functions, used in finding the derivative of ln(sin x).</li>
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</ul><ul><li><strong>Chain Rule:</strong>A rule for differentiating compositions of functions, used in finding the derivative of ln(sin x).</li>
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</ul><ul><li><strong>Cotangent Function:</strong>A trigonometric function that is the reciprocal of the tangent function, written as cot x.</li>
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</ul><ul><li><strong>Cotangent Function:</strong>A trigonometric function that is the reciprocal of the tangent function, written as cot x.</li>
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</ul><ul><li><strong>Undefined Points:</strong>Points at which a function is not defined, such as where sin(x) = 0 for ln(sin x).</li>
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</ul><ul><li><strong>Undefined Points:</strong>Points at which a function is not defined, such as where sin(x) = 0 for ln(sin x).</li>
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</ul><p>What Is Calculus? 🔢 | Easy Tricks, Limits & 🎯 Fun Learning for Kids | ✨BrightCHAMPS Math</p>
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</ul><p>What Is Calculus? 🔢 | Easy Tricks, Limits & 🎯 Fun Learning for Kids | ✨BrightCHAMPS Math</p>
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<p>▶</p>
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<p>▶</p>
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<h2>Jaskaran Singh Saluja</h2>
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<h2>Jaskaran Singh Saluja</h2>
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<h3>About the Author</h3>
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<h3>About the Author</h3>
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<p>Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.</p>
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<p>Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.</p>
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<h3>Fun Fact</h3>
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<h3>Fun Fact</h3>
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<p>: He loves to play the quiz with kids through algebra to make kids love it.</p>
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<p>: He loves to play the quiz with kids through algebra to make kids love it.</p>