Given the function y=(-x^(2)-5x)^(3), find (dy)/(dx) in any form. Answer: (dy)/(dx)=

Identify Functions: We are given the function y=(-x^2-5x)^3 and we need to find its derivative with respect to x. We will use the chain rule, which 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.Find Outer Function Derivative: First, let's identify the outer function and the inner function. The outer function is u^3 and the inner function is u = -x^2 - 5x. We will find the derivative of the outer function with respect to u, which is 3u^2.Find Inner Function Derivative: Now, we will find the derivative of the inner function with respect to x, which is the derivative of u = -x^2 - 5x. Using the power rule, the derivative of -x^2 is -2x, and the derivative of -5x is -5. So, the derivative of the inner function is du/dx = -2x - 5.Apply Chain Rule: Next, we apply the chain rule by multiplying the derivative of the outer function with respect to u by the derivative of the inner function with respect to x. This gives us dy/dx = 3u^2 * (-2x - 5).Substitute Back for Final Result: Substitute the inner function u back into the expression for dy/dx to get the derivative in terms of x. So, dy/dx = 3(-x^2 - 5x)^2 * (-2x - 5).

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Given the function
y=(-x^(2)-5x)^(3), find
(dy)/(dx) in any form.
Answer:
(dy)/(dx)=

Given the function $y=\left(-x^{2}-5 x\right)^{3}$ , find $\frac{d y}{d x}$ in any form. $\newline$ Answer: $\frac{d y}{d x}=$

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Algebra 2

Find the vertex of the transformed function

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Q. Given the function

y=\left(-x^{2}-5 x\right)^{3}

, find

\frac{d y}{d x}

in any form.

\newline

Answer:

\frac{d y}{d x}=

Identify Functions: We are given the function $y=(-x^2-5x)^3$ and we need to find its derivative with respect to $x$ . We will use the chain rule, which 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.
Find Outer Function Derivative: First, let's identify the outer function and the inner function. The outer function is $u^3$ and the inner function is $u = -x^2 - 5x$ . We will find the derivative of the outer function with respect to $u$ , which is $3u^2$ .
Find Inner Function Derivative: Now, we will find the derivative of the inner function with respect to $x$ , which is the derivative of $u = -x^2 - 5x$ . Using the power rule, the derivative of $-x^2$ is $-2x$ , and the derivative of $-5x$ is $-5$ . So, the derivative of the inner function is $\frac{du}{dx} = -2x - 5$ .
Apply Chain Rule: Next, we apply the chain rule by multiplying the derivative of the outer function with respect to $u$ by the derivative of the inner function with respect to $x$ . This gives us $\frac{dy}{dx} = 3u^2 \cdot (-2x - 5)$ .
Substitute Back for Final Result: Substitute the inner function $u$ back into the expression for $\frac{dy}{dx}$ to get the derivative in terms of $x$ . So, $\frac{dy}{dx} = 3(-x^2 - 5x)^2 \cdot (-2x - 5)$ .