Find (dy)/(dx), if y=(-5x^(2)-3)^(-4)

Identify function: Identify the function to differentiate. We are given the function y = (-5x^2 - 3)^(-4). We need to find the derivative of this function with respect to x.Apply chain rule: Apply the chain rule for differentiation. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. In this case, the outer function is u^(-4) and the inner function is u = -5x^2 - 3.Differentiate outer function: Differentiate the outer function with respect to the inner function u. The derivative of u^(-4) with respect to u is -4u^(-5).Differentiate inner function: Differentiate the inner function u with respect to x. The derivative of u = -5x^2 - 3 with respect to x is du/dx = -10x.Apply chain rule: Apply the chain rule by multiplying the derivatives from Step 3 and Step 4. (dy/dx) = (-4)(-5x^2 - 3)^(-5) * (-10x)Simplify expression: Simplify the expression. (dy/dx) = 40x(-5x^2 - 3)^(-5)

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Find $\frac{dy}{dx}$ , if $y=(-5x^{2}-3)^{-4}$

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Algebra 2

Sin, cos, and tan of special angles

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Q. Find

\frac{dy}{dx}

, if

y=(-5x^{2}-3)^{-4}

Identify function: Identify the function to differentiate. $\newline$ We are given the function $y = (-5x^2 - 3)^{-4}$ . We need to find the derivative of this function with respect to $x$ .
Apply chain rule: Apply the chain rule for differentiation. $\newline$ The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. In this case, the outer function is $u^{-4}$ and the inner function is $u = -5x^2 - 3$ .
Differentiate outer function: Differentiate the outer function with respect to the inner function $u$ . The derivative of $u^{-4}$ with respect to $u$ is $-4u^{-5}$ .
Differentiate inner function: Differentiate the inner function $u$ with respect to $x$ . The derivative of $u = -5x^2 - 3$ with respect to $x$ is $\frac{du}{dx} = -10x$ .
Apply chain rule: Apply the chain rule by multiplying the derivatives from Step $3$ and Step $4$ . $(\frac{dy}{dx}) = (-4)(-5x^2 - 3)^{-5} \times (-10x)$
Simplify expression: Simplify the expression. $(\frac{dy}{dx}) = 40x(-5x^2 - 3)^{-5}$