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Find the derivative of the following function.

y=9^(-x^(2))
Answer:
y^(')=

Find the derivative of the following function. $\newline$ $y=9^{-x^{2}}$ $\newline$ Answer: $y^{\prime}=$

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Find derivatives of radical functions

Full solution

Q. Find the derivative of the following function.

\newline

y=9^{-x^{2}}

\newline

Answer:

y^{\prime}=

Identify function: Identify the function to differentiate. $\newline$ We have $y = 9^{(-x^2)}$ . This is an exponential function with a base of $9$ and an exponent of $-x^2$ .
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. Here, the outer function is $9^u$ and the inner function is $u = -x^2$ .
Differentiate outer function: Differentiate the outer function with respect to the inner function $u$ . The derivative of $9^u$ with respect to $u$ is $9^u \cdot \ln(9)$ , where $\ln(9)$ is the natural logarithm of $9$ .
Differentiate inner function: Differentiate the inner function with respect to $x$ . The derivative of $u = -x^2$ with respect to $x$ is $-2x$ .
Apply chain rule: Apply the chain rule by multiplying the derivatives from steps $3$ and $4$ . $\newline$ The derivative of $y$ with respect to $x$ is $(9^{-x^2} \cdot \ln(9)) \cdot (-2x)$ .
Simplify expression: Simplify the expression. $\newline$ $y' = -2x \cdot 9^{-x^2} \cdot \ln(9)$ .

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