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Find the derivative of the following function.

y=e^(-5x^(3))
Answer:
y^(')=

Find the derivative of the following function. $\newline$ $y=e^{-5 x^{3}}$ $\newline$ 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^{3}}

\newline

Answer:

y^{\prime}=

Identify Function & Rule: Identify the function and the rule to use for differentiation. $\newline$ We have the function $y=e^{-5x^{3}}$ . 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. $\newline$ The outer function is $e^u$ where $u=-5x^{3}$ . The derivative of $e^u$ with respect to $u$ is $e^u$ . The inner function is $u=-5x^{3}$ . The derivative of $-5x^{3}$ with respect to $x$ is $-15x^{2}$ .
Multiply Derivatives: Multiply the derivatives of the outer and inner functions. $\newline$ Using the chain rule, we multiply the derivative of the outer function by the derivative of the inner function to get the derivative of the composite function. $\newline$ $y' = e^{-5x^{3}} \cdot (-15x^{2})$
Simplify Expression: Simplify the expression if possible. $\newline$ The expression for the derivative is already simplified, so we can state the final answer. $\newline$ $y' = -15x^{2}e^{-5x^{3}}$

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