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$y=-(x+7)^2-9$

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Find derivatives using the chain rule II

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Q.

y=-(x+7)^2-9

Identify function: Identify the function to differentiate. $\newline$ We are given the function $y = -(x+7)^2 - 9$ , and we need to find its derivative 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 $-u^2$ and the inner function is $u = x + 7$ .
Differentiate outer function: Differentiate the outer function with respect to the inner function $u$ . The outer function is $-u^2$ . Its derivative with respect to $u$ is $-2u$ .
Differentiate inner function: Differentiate the inner function with respect to $x$ . The inner function is $u = x + 7$ . Its derivative with respect to $x$ is $1$ , since the derivative of $x$ is $1$ and the derivative of a constant is $0$ .
Apply chain rule multiplication: Apply the chain rule by multiplying the derivatives from Step $3$ and Step $4$ . $\newline$ The derivative of $y$ with respect to $x$ is the product of the derivative of the outer function and the derivative of the inner function: $\frac{dy}{dx} = -2u \times 1$ .
Substitute inner function: Substitute the inner function back into the derivative. $\newline$ Replace $u$ with $x + 7$ in the derivative we found in Step $5$ : $\frac{dy}{dx} = -2(x + 7)$ .
Simplify derivative: Simplify the derivative. $\newline$ The derivative simplifies to $\frac{dy}{dx} = -2x - 14$ .

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