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Find the derivative of the following function.

y=e^(7x^(5))
Answer:
y^(')=

Find the derivative of the following function. $\newline$ $y=e^{7 x^{5}}$ $\newline$ 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^{7 x^{5}}

\newline

Answer:

y^{\prime}=

Identify Function & Type: Identify the function and the type of differentiation required. $\newline$ We are given the function $y = e^{7x^{5}}$ and we need to find its derivative with respect to $x$ . This is a case of finding the derivative of an exponential function with a composite function $(7x^{5})$ as the exponent.
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 $u = 7x^{5}$ ) and the inner function is $7x^{5}$ .
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 $e^{7x^{5}}$ with respect to $7x^{5}$ is $e^{7x^{5}}$ .
Differentiate Inner Function: Differentiate the inner function with respect to $x$ . The inner function is $7x^{5}$ . Using the power rule, the derivative of $x^{n}$ with respect to $x$ is $n\cdot x^{(n-1)}$ , so the derivative of $7x^{5}$ with respect to $x$ is $35x^{4}$ .
Multiply Derivatives: Multiply the derivatives from Step $3$ and Step $4$ . $\newline$ We multiply the derivative of the outer function by the derivative of the inner function to get the derivative of the composite function. Therefore, $y' = e^{7x^{5}} \times 35x^{4}$ .

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