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2026-01-01
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<p>Last updated on<strong>August 5, 2025</strong></p>
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<p>Last updated on<strong>August 5, 2025</strong></p>
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<p>We use the derivative of 9x, which is 9, as a measuring tool for how the linear function changes in response to a slight change in x. Derivatives help us calculate profit or loss in real-life situations. We will now talk about the derivative of 9x in detail.</p>
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<p>We use the derivative of 9x, which is 9, as a measuring tool for how the linear function changes in response to a slight change in x. Derivatives help us calculate profit or loss in real-life situations. We will now talk about the derivative of 9x in detail.</p>
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<h2>What is the Derivative of 9x?</h2>
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<h2>What is the Derivative of 9x?</h2>
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<p>We now understand the derivative<a>of</a>9x. It is commonly represented as d/dx (9x) or (9x)', and its value is 9. </p>
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<p>We now understand the derivative<a>of</a>9x. It is commonly represented as d/dx (9x) or (9x)', and its value is 9. </p>
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<p>The<a>function</a>9x has a clearly defined derivative, indicating it is differentiable across all<a>real numbers</a>.</p>
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<p>The<a>function</a>9x has a clearly defined derivative, indicating it is differentiable across all<a>real numbers</a>.</p>
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<p>The key concepts are mentioned below: Linear Function: A function of the form f(x) = ax + b.</p>
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<p>The key concepts are mentioned below: Linear Function: A function of the form f(x) = ax + b.</p>
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<p>Constant Rule: The derivative of a<a>constant</a>is zero.</p>
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<p>Constant Rule: The derivative of a<a>constant</a>is zero.</p>
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<p>Coefficient Rule: The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.</p>
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<p>Coefficient Rule: The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.</p>
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<h2>Derivative of 9x Formula</h2>
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<h2>Derivative of 9x Formula</h2>
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<p>The derivative of 9x can be denoted as d/dx (9x) or (9x)'. </p>
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<p>The derivative of 9x can be denoted as d/dx (9x) or (9x)'. </p>
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<p>The<a>formula</a>we use to differentiate 9x is: d/dx (9x) = 9 The formula applies to all values of x.</p>
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<p>The<a>formula</a>we use to differentiate 9x is: d/dx (9x) = 9 The formula applies to all values of x.</p>
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<h2>Proofs of the Derivative of 9x</h2>
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<h2>Proofs of the Derivative of 9x</h2>
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<p>We can derive the derivative of 9x using basic<a>calculus</a>rules. To show this, we will use the fundamental rules of differentiation. </p>
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<p>We can derive the derivative of 9x using basic<a>calculus</a>rules. To show this, we will use the fundamental rules of differentiation. </p>
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<p>There are several methods we use to prove this, such as:</p>
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<p>There are several methods we use to prove this, such as:</p>
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<p>By Definition Using Constant Rule Using Coefficient Rule We will now demonstrate that the differentiation of 9x results in 9 using the above-mentioned methods:</p>
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<p>By Definition Using Constant Rule Using Coefficient Rule We will now demonstrate that the differentiation of 9x results in 9 using the above-mentioned methods:</p>
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<p>By Definition The derivative of 9x can be proved using the definition of the derivative, which expresses the derivative as the limit of the difference<a>quotient</a>. To find the derivative of 9x using the definition, we will consider f(x) = 9x.</p>
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<p>By Definition The derivative of 9x can be proved using the definition of the derivative, which expresses the derivative as the limit of the difference<a>quotient</a>. To find the derivative of 9x using the definition, we will consider f(x) = 9x.</p>
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<p>Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1) Given that f(x) = 9x, we write f(x + h) = 9(x + h).</p>
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<p>Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1) Given that f(x) = 9x, we write f(x + h) = 9(x + h).</p>
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<p>Substituting these into<a>equation</a>(1), f'(x) = limₕ→₀ [9(x + h) - 9x] / h = limₕ→₀ [9x + 9h - 9x] / h = limₕ→₀ [9h] / h = limₕ→₀ 9 = 9 Hence, proved. Using Constant Rule To prove the differentiation of 9x using the constant rule, We use the formula: d/dx (c*f(x)) = c * f'(x) Here, c = 9 and f(x) = x Since the derivative of f(x) = x is 1, d/dx (9x) = 9 * 1 = 9</p>
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<p>Substituting these into<a>equation</a>(1), f'(x) = limₕ→₀ [9(x + h) - 9x] / h = limₕ→₀ [9x + 9h - 9x] / h = limₕ→₀ [9h] / h = limₕ→₀ 9 = 9 Hence, proved. Using Constant Rule To prove the differentiation of 9x using the constant rule, We use the formula: d/dx (c*f(x)) = c * f'(x) Here, c = 9 and f(x) = x Since the derivative of f(x) = x is 1, d/dx (9x) = 9 * 1 = 9</p>
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<p>Using Coefficient Rule We will now prove the derivative of 9x using the<a>coefficient</a>rule.</p>
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<p>Using Coefficient Rule We will now prove the derivative of 9x using the<a>coefficient</a>rule.</p>
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<p>The step-by-step process is demonstrated below: Given that, u(x) = x and c = 9</p>
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<p>The step-by-step process is demonstrated below: Given that, u(x) = x and c = 9</p>
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<p>Using the coefficient rule formula: d/dx [c*u(x)] = c * u'(x) u'(x) = d/dx (x) = 1 Thus, d/dx (9x) = 9 * 1 = 9.</p>
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<p>Using the coefficient rule formula: d/dx [c*u(x)] = c * u'(x) u'(x) = d/dx (x) = 1 Thus, d/dx (9x) = 9 * 1 = 9.</p>
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<p>Hence, proved.</p>
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<p>Hence, proved.</p>
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<h2>Higher-Order Derivatives of 9x</h2>
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<h2>Higher-Order Derivatives of 9x</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.</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.</p>
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<p>Higher-order derivatives make it easier to understand functions like 9x. For the first derivative of a function, we write f′(x), which indicates how the function changes or its slope at a certain point.</p>
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<p>Higher-order derivatives make it easier to understand functions like 9x. For the first derivative of a function, we write f′(x), which indicates how the function changes or its slope at a certain point.</p>
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<p>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>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 9x, 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 9x, 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>The function 9x is a linear function and does not have special cases of undefined points or asymptotes. For any value of x, the derivative of 9x remains 9.</p>
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<p>The function 9x is a linear function and does not have special cases of undefined points or asymptotes. For any value of x, the derivative of 9x remains 9.</p>
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<h2>Common Mistakes and How to Avoid Them in Derivatives of 9x</h2>
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<h2>Common Mistakes and How to Avoid Them in Derivatives of 9x</h2>
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<p>Students frequently make mistakes when differentiating 9x. </p>
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<p>Students frequently make mistakes when differentiating 9x. </p>
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<p>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>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 (9x · 5x).</p>
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<p>Calculate the derivative of (9x · 5x).</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) = 9x · 5x. Using the product rule, f'(x) = u′v + uv′ In the given equation, u = 9x and v = 5x. Let’s differentiate each term, u′ = d/dx (9x) = 9 v′ = d/dx (5x) = 5</p>
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<p>Here, we have f(x) = 9x · 5x. Using the product rule, f'(x) = u′v + uv′ In the given equation, u = 9x and v = 5x. Let’s differentiate each term, u′ = d/dx (9x) = 9 v′ = d/dx (5x) = 5</p>
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<p>Substitute these into the given equation, f'(x) = (9) · (5x) + (9x) · (5)</p>
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<p>Substitute these into the given equation, f'(x) = (9) · (5x) + (9x) · (5)</p>
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<p>Simplifying terms gives us the final answer, f'(x) = 45x + 45. Thus, the derivative of the specified function is 45x + 45.</p>
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<p>Simplifying terms gives us the final answer, f'(x) = 45x + 45. Thus, the derivative of the specified function is 45x + 45.</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. 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>We find the derivative of the given function by dividing the function into two parts. 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 company calculates the cost of production using the function y = 9x, where y represents the cost for producing x items. If x = 100 items, determine the rate of change of cost.</p>
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<p>A company calculates the cost of production using the function y = 9x, where y represents the cost for producing x items. If x = 100 items, determine the rate of change of cost.</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 = 9x (cost function)...(1) Now, we will differentiate the equation (1) Take the derivative: dy/dx = 9</p>
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<p>We have y = 9x (cost function)...(1) Now, we will differentiate the equation (1) Take the derivative: dy/dx = 9</p>
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<p>Given x = 100 (substitute this into the derivative)</p>
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<p>Given x = 100 (substitute this into the derivative)</p>
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<p>The rate of change of cost remains constant at 9, regardless of the value of x.</p>
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<p>The rate of change of cost remains constant at 9, regardless of the value of x.</p>
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<p>Hence, the rate of change of cost for producing 100 items is 9.</p>
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<p>Hence, the rate of change of cost for producing 100 items is 9.</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 cost using the derivative of the cost function, which remains constant at 9 for any number of items produced.</p>
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<p>We find the rate of change of cost using the derivative of the cost function, which remains constant at 9 for any number of items produced.</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 = 9x.</p>
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<p>Derive the second derivative of the function y = 9x.</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 = 9...(1) Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [9] Since the derivative of a constant is 0, d²y/dx² = 0. Therefore, the second derivative of the function y = 9x is 0.</p>
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<p>The first step is to find the first derivative, dy/dx = 9...(1) Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [9] Since the derivative of a constant is 0, d²y/dx² = 0. Therefore, the second derivative of the function y = 9x is 0.</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. Since the first derivative is a constant, the second derivative is zero.</p>
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<p>We use the step-by-step process, where we start with the first derivative. Since the first derivative is a constant, the second derivative is zero.</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 (9x²) = 18x.</p>
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<p>Prove: d/dx (9x²) = 18x.</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 power rule: Consider y = 9x² To differentiate, we use the power rule: dy/dx = 9 * d/dx [x²] Since the derivative of x² is 2x, dy/dx = 9 * 2x Therefore, d/dx (9x²) = 18x. Hence proved.</p>
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<p>Let’s start using the power rule: Consider y = 9x² To differentiate, we use the power rule: dy/dx = 9 * d/dx [x²] Since the derivative of x² is 2x, dy/dx = 9 * 2x Therefore, d/dx (9x²) = 18x. 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 power rule to differentiate the equation. Then, we replace x² with its derivative. As a final step, we substitute and simplify to derive the equation.</p>
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<p>In this step-by-step process, we used the power rule to differentiate the equation. Then, we replace x² with its derivative. As a final step, we substitute and 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 (9x/x).</p>
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<p>Solve: d/dx (9x/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 first simplify it: d/dx (9x/x) = d/dx (9) Since 9 is a constant, its derivative is 0. Therefore, d/dx (9x/x) = 0.</p>
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<p>To differentiate the function, we first simplify it: d/dx (9x/x) = d/dx (9) Since 9 is a constant, its derivative is 0. Therefore, d/dx (9x/x) = 0.</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>In this process, we simplify the given function and use the differentiation rule for constants to obtain the final result.</p>
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<p>In this process, we simplify the given function and use the differentiation rule for constants 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 9x</h2>
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<h2>FAQs on the Derivative of 9x</h2>
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<h3>1.Find the derivative of 9x.</h3>
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<h3>1.Find the derivative of 9x.</h3>
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<p>The derivative of 9x is a constant value of 9, as it is a linear function.</p>
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<p>The derivative of 9x is a constant value of 9, as it is a linear function.</p>
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<h3>2.Can we use the derivative of 9x in real life?</h3>
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<h3>2.Can we use the derivative of 9x in real life?</h3>
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<p>Yes, we can use the derivative of 9x in real life to calculate constant rates of change, such as cost per item or speed per hour.</p>
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<p>Yes, we can use the derivative of 9x in real life to calculate constant rates of change, such as cost per item or speed per hour.</p>
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<h3>3.Is it possible to take the derivative of 9x at any point?</h3>
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<h3>3.Is it possible to take the derivative of 9x at any point?</h3>
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<p>Yes, it is possible to take the derivative of 9x at any point, as it is defined across all real<a>numbers</a>and has a constant derivative of 9.</p>
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<p>Yes, it is possible to take the derivative of 9x at any point, as it is defined across all real<a>numbers</a>and has a constant derivative of 9.</p>
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<h3>4.What rule is used to differentiate 9x/x?</h3>
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<h3>4.What rule is used to differentiate 9x/x?</h3>
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<p>To differentiate 9x/x, simplify it to 9 and use the constant rule, which states the derivative of a constant is 0.</p>
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<p>To differentiate 9x/x, simplify it to 9 and use the constant rule, which states the derivative of a constant is 0.</p>
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<h3>5.Are the derivatives of 9x and x⁹ the same?</h3>
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<h3>5.Are the derivatives of 9x and x⁹ the same?</h3>
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<p>No, they are different. The derivative of 9x is 9, while the derivative of x⁹ is 9x⁸.</p>
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<p>No, they are different. The derivative of 9x is 9, while the derivative of x⁹ is 9x⁸.</p>
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<h2>Important Glossaries for the Derivative of 9x</h2>
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<h2>Important Glossaries for the Derivative of 9x</h2>
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<ul><li>Derivative: 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>Derivative: 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>Linear Function: A function of the form f(x) = ax + b, where the graph is a straight line.</li>
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</ul><ul><li>Linear Function: A function of the form f(x) = ax + b, where the graph is a straight line.</li>
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</ul><ul><li>Constant Rule: A rule stating that the derivative of a constant is zero.</li>
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</ul><ul><li>Constant Rule: A rule stating that the derivative of a constant is zero.</li>
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</ul><ul><li>Coefficient Rule: A rule stating that the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.</li>
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</ul><ul><li>Coefficient Rule: A rule stating that the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.</li>
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</ul><ul><li>Power Rule: A basic differentiation rule used to find the derivative of a power of x, such as xⁿ.</li>
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</ul><ul><li>Power Rule: A basic differentiation rule used to find the derivative of a power of x, such as xⁿ.</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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<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>