Given the function y=-(5x^(2)+2x+1)^(4), find (dy)/(dx) in any form. Answer: (dy)/(dx)=

Identify Function: Identify the function to differentiate. We are given the function y = -(5x^(2) + 2x + 1)^(4) and we need to find the derivative of this function with respect to x, which is denoted as (dy)/(dx).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) + 2x + 1).Differentiate Outer Function: Differentiate the outer function with respect to the inner function. Let u = -(5x^(2) + 2x + 1). The derivative of u^(4) with respect to u is 4u^(3).Differentiate Inner Function: Differentiate the inner function with respect to x. The derivative of u = -(5x^(2) + 2x + 1) with respect to x is du/dx = -[2(5x) + 2] = -10x - 2.Apply Chain Rule: Apply the chain rule by multiplying the derivatives from Step 3 and Step 4. (dy)/(dx) = 4u^(3) * (du/dx) = 4[-(5x^(2) + 2x + 1)]^(3) * (-10x - 2).Simplify Derivative: Simplify the expression for the derivative. (dy)/(dx) = 4[-(5x^(2) + 2x + 1)]^(3) * (-10x - 2) = -4(5x^(2) + 2x + 1)^(3) * (10x + 2).

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Given the function
y=-(5x^(2)+2x+1)^(4), find
(dy)/(dx) in any form.
Answer:
(dy)/(dx)=

Given the function $y=-\left(5 x^{2}+2 x+1\right)^{4}$ , find $\frac{d y}{d x}$ in any form. $\newline$ Answer: $\frac{d y}{d x}=$

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Algebra 2

Evaluate exponential functions

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Q. Given the function

y=-\left(5 x^{2}+2 x+1\right)^{4}

, find

\frac{d y}{d x}

in any form.

\newline

Answer:

\frac{d y}{d x}=

Identify Function: Identify the function to differentiate. $\newline$ We are given the function $y = -(5x^{2} + 2x + 1)^{4}$ and we need to find the derivative of this function with respect to $x$ , which is denoted as $\frac{dy}{dx}$ .
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 $u = -(5x^{2} + 2x + 1)$ .
Differentiate Outer Function: Differentiate the outer function with respect to the inner function. $\newline$ Let $u = -(5x^{2} + 2x + 1)$ . The derivative of $u^{4}$ with respect to $u$ is $4u^{3}$ .
Differentiate Inner Function: Differentiate the inner function with respect to $x$ . The derivative of $u = -(5x^{2} + 2x + 1)$ with respect to $x$ is $\frac{du}{dx} = -[2(5x) + 2] = -10x - 2$ .
Apply Chain Rule: Apply the chain rule by multiplying the derivatives from Step $3$ and Step $4$ . $\newline$ $(\frac{dy}{dx}) = 4u^{3} \cdot (\frac{du}{dx}) = 4[-(5x^{2} + 2x + 1)]^{3} \cdot (-10x - 2)$ .
Simplify Derivative: Simplify the expression for the derivative. $\newline$ $\frac{dy}{dx} = 4[-(5x^{2} + 2x + 1)]^{3} * (-10x - 2) = -4(5x^{2} + 2x + 1)^{3} * (10x + 2)$ .