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Find the derivative of the following function.

y=4^(2x^(5))
Answer:
y^(')=

Find the derivative of the following function. $\newline$ $y=4^{2 x^{5}}$ $\newline$ Answer: $y^{\prime}=$

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Find derivatives using logarithmic differentiation

Full solution

Q. Find the derivative of the following function.

\newline

y=4^{2 x^{5}}

\newline

Answer:

y^{\prime}=

Recognize composition of functions: First, we need to recognize that this is a composition of functions and we will need to use the chain rule to find the derivative. 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.
Find derivative of outer function: The outer function is $4^u$ where $u = 2x^5$ . The derivative of $4^u$ with respect to $u$ is $4^u \cdot \ln(4)$ because the derivative of $a^u$ is $a^u \cdot \ln(a)$ where $a$ is a constant.
Find derivative of inner function: The inner function is $u = 2x^5$ . The derivative of $u$ with respect to $x$ is $10x^4$ because the derivative of $x^n$ is $n\cdot x^{(n-1)}$ .
Apply chain rule: Now we apply the chain rule by multiplying the derivative of the outer function by the derivative of the inner function. This gives us: $\newline$ $y' = (4^{2x^5} \cdot \ln(4)) \cdot (10x^4)$
Simplify final answer: Simplify the expression to get the final answer: $\newline$ $y' = 10x^4 \cdot 4^{(2x^5)} \cdot \ln(4)$

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