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Find the derivative of the following function.

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

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

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Find derivatives using logarithmic differentiation

Full solution

Q. Find the derivative of the following function.

\newline

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

\newline

Answer:

y^{\prime}=

Identify Function: Identify the function to differentiate. $\newline$ We are given the function $y = e^{x^2 - 9x}$ . 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 $e^{u}$ , where $u = x^{2} - 9x$ , and the inner function is $u(x) = x^{2} - 9x$ .
Differentiate Outer Function: Differentiate the outer function with respect to the inner function. $\newline$ The derivative of $e^u$ with respect to $u$ is $e^u$ . So, the derivative of $e^{x^2 - 9x}$ with respect to $x^2 - 9x$ is $e^{x^2 - 9x}$ .
Differentiate Inner Function: Differentiate the inner function with respect to $x$ . The derivative of $u(x) = x^2 - 9x$ with respect to $x$ is $2x - 9$ .
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. Therefore, $y' = e^{(x^2 - 9x)} \cdot (2x - 9)$ .

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