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2026-01-01
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<p>Last updated on<strong>September 9, 2025</strong></p>
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<p>Last updated on<strong>September 9, 2025</strong></p>
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<p>We use the derivative of -x², which is -2x, as a tool for understanding how the 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 -x² in detail.</p>
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<p>We use the derivative of -x², which is -2x, as a tool for understanding how the 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 -x² in detail.</p>
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<h2>What is the Derivative of -x²?</h2>
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<h2>What is the Derivative of -x²?</h2>
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<p>We now understand the derivative<a>of</a>-x².</p>
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<p>We now understand the derivative<a>of</a>-x².</p>
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<p>It is commonly represented as d/dx (-x²) or (-x²)', and its value is -2x.</p>
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<p>It is commonly represented as d/dx (-x²) or (-x²)', and its value is -2x.</p>
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<p>The<a>function</a>-x² has a clearly defined derivative, indicating it is differentiable within its domain.</p>
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<p>The<a>function</a>-x² 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>Polynomial Function: A function like -x², which is a basic<a>polynomial</a>.</p>
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<p>Polynomial Function: A function like -x², which is a basic<a>polynomial</a>.</p>
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<p>Power Rule: A rule used for differentiating polynomial functions.</p>
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<p>Power Rule: A rule used for differentiating polynomial functions.</p>
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<h2>Derivative of -x² Formula</h2>
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<h2>Derivative of -x² Formula</h2>
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<p>The derivative of -x² can be denoted as d/dx (-x²) or (-x²)'. The<a>formula</a>we use to differentiate -x² is: d/dx (-x²) = -2x The formula applies to all x in the<a>real number</a>domain.</p>
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<p>The derivative of -x² can be denoted as d/dx (-x²) or (-x²)'. The<a>formula</a>we use to differentiate -x² is: d/dx (-x²) = -2x The formula applies to all x in the<a>real number</a>domain.</p>
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<h2>Proofs of the Derivative of -x²</h2>
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<h2>Proofs of the Derivative of -x²</h2>
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<p>We can derive the derivative of -x² using proofs.</p>
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<p>We can derive the derivative of -x² using proofs.</p>
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<p>To show this, we will use the rules of differentiation.</p>
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<p>To show this, we will use the 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>Using the Power Rule</p>
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<p>Using the Power Rule</p>
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<p>Using the First Principle</p>
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<p>Using the First Principle</p>
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<p>Using the Power Rule</p>
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<p>Using the Power Rule</p>
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<p>To prove the differentiation of -x² using the<a>power</a>rule,</p>
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<p>To prove the differentiation of -x² using the<a>power</a>rule,</p>
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<p>We use the formula: d/dx (x^n) = n*x^(n-1)</p>
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<p>We use the formula: d/dx (x^n) = n*x^(n-1)</p>
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<p>For our function, n = 2 and the<a>coefficient</a>is -1: d/dx (-x²) = -1 * 2 * x^(2-1) = -2x</p>
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<p>For our function, n = 2 and the<a>coefficient</a>is -1: d/dx (-x²) = -1 * 2 * x^(2-1) = -2x</p>
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<p>Hence, proved.</p>
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<p>Hence, proved.</p>
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<p>Using the First Principle</p>
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<p>Using the First Principle</p>
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<p>The derivative of -x² can be proved using the First Principle, which expresses the derivative as the limit of the difference<a>quotient</a>.</p>
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<p>The derivative of -x² can be proved using the First Principle, which expresses the derivative as the limit of the difference<a>quotient</a>.</p>
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<p>To find the derivative of -x² using the first principle, we will consider f(x) = -x².</p>
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<p>To find the derivative of -x² using the first principle, we will consider f(x) = -x².</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)</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)</p>
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<p>Given that f(x) = -x², we write f(x + h) = -(x + h)².</p>
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<p>Given that f(x) = -x², we write f(x + h) = -(x + h)².</p>
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<p>Substituting these into<a>equation</a>(1), f'(x) = limₕ→₀ [-(x + h)² + x²] / h = limₕ→₀ [-(x² + 2xh + h²) + x²] / h = limₕ→₀ [-2xh - h²] / h = limₕ→₀ [-2x - h] As h approaches 0, the<a>expression</a>simplifies to: f'(x) = -2x</p>
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<p>Substituting these into<a>equation</a>(1), f'(x) = limₕ→₀ [-(x + h)² + x²] / h = limₕ→₀ [-(x² + 2xh + h²) + x²] / h = limₕ→₀ [-2xh - h²] / h = limₕ→₀ [-2x - h] As h approaches 0, the<a>expression</a>simplifies to: f'(x) = -2x</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 -x²</h2>
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<h2>Higher-Order Derivatives of -x²</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 -x².</p>
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<p>Higher-order derivatives make it easier to understand functions like -x².</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.</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.</p>
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<p>The second derivative is derived from the first derivative, which is denoted using f′′(x).</p>
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<p>The second derivative is derived from the first derivative, which is denoted using f′′(x).</p>
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<p>Similarly, the third derivative, f′′′(x), is the result of the second derivative and this pattern continues.</p>
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<p>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 -x², 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 -x², 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 of -x² is -2(0), which is 0.</p>
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<p>When x is 0, the derivative of -x² is -2(0), which is 0.</p>
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<h2>Common Mistakes and How to Avoid Them in Derivatives of -x²</h2>
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<h2>Common Mistakes and How to Avoid Them in Derivatives of -x²</h2>
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<p>Students frequently make mistakes when differentiating -x². 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 -x². 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 (-x² · x³)</p>
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<p>Calculate the derivative of (-x² · x³)</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) = -x² · x³.</p>
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<p>Here, we have f(x) = -x² · x³.</p>
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<p>Using the product rule, f'(x) = u′v + uv′</p>
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<p>Using the product rule, f'(x) = u′v + uv′</p>
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<p>In the given equation, u = -x² and v = x³.</p>
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<p>In the given equation, u = -x² and v = x³.</p>
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<p>Let’s differentiate each term, u′ = d/dx (-x²) = -2x v′ = d/dx (x³) = 3x² substituting into the given equation, f'(x) = (-2x)(x³) + (-x²)(3x²)</p>
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<p>Let’s differentiate each term, u′ = d/dx (-x²) = -2x v′ = d/dx (x³) = 3x² substituting into the given equation, f'(x) = (-2x)(x³) + (-x²)(3x²)</p>
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<p>Let’s simplify terms to get the final answer, f'(x) = -2x⁴ - 3x⁴ f'(x) = -5x⁴</p>
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<p>Let’s simplify terms to get the final answer, f'(x) = -2x⁴ - 3x⁴ f'(x) = -5x⁴</p>
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<p>Thus, the derivative of the specified function is -5x⁴.</p>
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<p>Thus, the derivative of the specified function is -5x⁴.</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 car's speed reduction is represented by the function y = -x², where y represents the deceleration at a distance x. If x = 3 meters, calculate the rate of change of deceleration.</p>
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<p>A car's speed reduction is represented by the function y = -x², where y represents the deceleration at a distance x. If x = 3 meters, calculate the rate of change of deceleration.</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 = -x² (deceleration of the car)...(1)</p>
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<p>We have y = -x² (deceleration of the car)...(1)</p>
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<p>Now, we will differentiate the equation (1)</p>
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<p>Now, we will differentiate the equation (1)</p>
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<p>Take the derivative -x²: dy/dx = -2x</p>
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<p>Take the derivative -x²: dy/dx = -2x</p>
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<p>Given x = 3 (substitute this into the derivative) dy/dx = -2(3) = -6</p>
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<p>Given x = 3 (substitute this into the derivative) dy/dx = -2(3) = -6</p>
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<p>Hence, the rate of change of deceleration at a distance x = 3 meters is -6.</p>
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<p>Hence, the rate of change of deceleration at a distance x = 3 meters is -6.</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 deceleration at x = 3 meters as -6, which means that at this point, the car's speed is decreasing at a rate of 6 units per meter.</p>
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<p>We find the rate of change of deceleration at x = 3 meters as -6, which means that at this point, the car's speed is decreasing at a rate of 6 units per meter.</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 = -x².</p>
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<p>Derive the second derivative of the function y = -x².</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 = -2x...(1)</p>
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<p>The first step is to find the first derivative, dy/dx = -2x...(1)</p>
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<p>Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx[-2x] d²y/dx² = -2</p>
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<p>Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx[-2x] d²y/dx² = -2</p>
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<p>Therefore, the second derivative of the function y = -x² is -2.</p>
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<p>Therefore, the second derivative of the function y = -x² is -2.</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. By differentiating -2x, we find that the second derivative is a constant value, -2.</p>
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<p>We use the step-by-step process, where we start with the first derivative. By differentiating -2x, we find that the second derivative is a constant value, -2.</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 (-(x²)²) = -4x³.</p>
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<p>Prove: d/dx (-(x²)²) = -4x³.</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:</p>
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<p>Let’s start using the chain rule:</p>
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<p>Consider y = -(x²)² y = -(x⁴)</p>
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<p>Consider y = -(x²)² y = -(x⁴)</p>
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<p>To differentiate, we use the power rule: dy/dx = -1 * 4x³ dy/dx = -4x³</p>
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<p>To differentiate, we use the power rule: dy/dx = -1 * 4x³ dy/dx = -4x³</p>
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<p>Hence proved.</p>
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<p>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. We then simplify to find the derivative of the given function.</p>
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<p>In this step-by-step process, we used the power rule to differentiate the equation. We then simplify to find the derivative of the given function.</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 (-x²/x)</p>
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<p>Solve: d/dx (-x²/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 (-x²/x) = (d/dx (-x²) * x - (-x²) * d/dx(x))/ x²</p>
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<p>To differentiate the function, we use the quotient rule: d/dx (-x²/x) = (d/dx (-x²) * x - (-x²) * d/dx(x))/ x²</p>
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<p>We will substitute d/dx (-x²) = -2x and d/dx (x) = 1 = (-2x * x - (-x²) * 1) / x² = (-2x² + x²) / x² = -x² / x² = -1</p>
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<p>We will substitute d/dx (-x²) = -2x and d/dx (x) = 1 = (-2x * x - (-x²) * 1) / x² = (-2x² + x²) / x² = -x² / x² = -1</p>
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<p>Therefore, d/dx (-x²/x) = -1</p>
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<p>Therefore, d/dx (-x²/x) = -1</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. As a final step, we simplify the equation to obtain the final result.</p>
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<p>In this process, we differentiate the given function using the quotient rule. 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 -x²</h2>
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<h2>FAQs on the Derivative of -x²</h2>
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<h3>1.Find the derivative of -x².</h3>
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<h3>1.Find the derivative of -x².</h3>
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<p>Using the power rule for -x², d/dx (-x²) = -2x</p>
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<p>Using the power rule for -x², d/dx (-x²) = -2x</p>
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<h3>2.Can we use the derivative of -x² in real life?</h3>
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<h3>2.Can we use the derivative of -x² in real life?</h3>
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<p>Yes, we can use the derivative of -x² in real life to understand rates of change in various fields, such as physics to calculate deceleration or in economics for cost functions.</p>
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<p>Yes, we can use the derivative of -x² in real life to understand rates of change in various fields, such as physics to calculate deceleration or in economics for cost functions.</p>
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<h3>3.What happens when x is 0 for the derivative of -x²?</h3>
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<h3>3.What happens when x is 0 for the derivative of -x²?</h3>
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<p>When x is 0, the derivative of -x² is -2(0), which equals 0.</p>
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<p>When x is 0, the derivative of -x² is -2(0), which equals 0.</p>
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<h3>4.What rule is used to differentiate -x²/x?</h3>
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<h3>4.What rule is used to differentiate -x²/x?</h3>
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<p>We use the quotient rule to differentiate -x²/x, d/dx (-x²/x) = (-2x * x - (-x²) * 1) / x² = -1.</p>
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<p>We use the quotient rule to differentiate -x²/x, d/dx (-x²/x) = (-2x * x - (-x²) * 1) / x² = -1.</p>
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<h3>5.Are the derivatives of -x² and -2x the same?</h3>
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<h3>5.Are the derivatives of -x² and -2x the same?</h3>
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<p>No, they are different. The derivative of -x² is -2x, while -2x is already a derivative of some linear function.</p>
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<p>No, they are different. The derivative of -x² is -2x, while -2x is already a derivative of some linear function.</p>
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<h2>Important Glossaries for the Derivative of -x²</h2>
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<h2>Important Glossaries for the Derivative of -x²</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>Polynomial Function:</strong>A mathematical expression involving a sum of powers in one or more variables multiplied by coefficients.</li>
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</ul><ul><li><strong>Polynomial Function:</strong>A mathematical expression involving a sum of powers in one or more variables multiplied by coefficients.</li>
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</ul><ul><li><strong>Power Rule:</strong>A basic rule in calculus used to find the derivative of a power of x.</li>
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</ul><ul><li><strong>Power Rule:</strong>A basic rule in calculus used to find the derivative of a power of x.</li>
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</ul><ul><li><strong>First Principle:</strong>A fundamental method used to define the derivative of a function.</li>
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</ul><ul><li><strong>First Principle:</strong>A fundamental method used to define the derivative of a function.</li>
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</ul><ul><li><strong>Quotient Rule:</strong>A rule for finding the derivative of a quotient of two functions.</li>
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</ul><ul><li><strong>Quotient Rule:</strong>A rule for finding the derivative of a quotient of two functions.</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>