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Find the derivative of the following function.

y=e^(-5x^(5))
Answer:
y^(')=

Find the derivative of the following function. $\newline$ $y=e^{-5 x^{5}}$ $\newline$ 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^{-5 x^{5}}

\newline

Answer:

y^{\prime}=

Identify Function and Rule: Identify the function and the rule to use for differentiation. $\newline$ We have the function $y = e^{-5x^{5}}$ . To find the derivative $y'$ , 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.
Apply Chain Rule: Apply the chain rule to differentiate the function. $\newline$ The outer function is $e^u$ where $u = -5x^{5}$ . The derivative of $e^u$ with respect to $u$ is $e^u$ . The inner function is $u = -5x^{5}$ , and its derivative with respect to $x$ is $-5 \times 5x^{5-1} = -25x^{4}$ .
Multiply Derivatives: Multiply the derivatives of the outer and inner functions. $\newline$ $y' = e^{-5x^{5}} \times (-25x^{4})$
Simplify Expression: Simplify the expression if necessary. $\newline$ The expression is already simplified, so we have the final answer. $\newline$ $y' = -25x^{4}e^{-5x^{5}}$

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