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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 & Rule: Identify the function and the rule to use for differentiation. $\newline$ We have the function $y=e^{5x^{5}}$ . To find the derivative, 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. 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 $u=5x^{5}$ . The derivative of $u$ with respect to $x$ is $5\cdot(5x^{5-1})=25x^{4}$ . Now, we multiply the derivative of the outer function by the derivative of the inner function. $y' = e^{5x^{5}} \cdot 25x^{4}$
Simplify Derivative: Simplify the expression for the derivative. $\newline$ $y' = 25x^{4} \cdot e^{5x^{5}}$ $\newline$ This is the simplified form of the derivative.

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