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Find the derivative of the following function.

y=4^(7x^(4))
Answer:
y^(')=

Find the derivative of the following function. $\newline$ $y=4^{7 x^{4}}$ $\newline$ 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^{7 x^{4}}

\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.
Derivative of Outer Function: The outer function is $4^u$ where $u=7x^4$ . The derivative of $4^u$ with respect to $u$ is $ext{ln}(4)*4^u$ because the derivative of $a^u$ is $ext{ln}(a)*a^u$ .
Derivative of Inner Function: The inner function is $u=7x^4$ . The derivative of $u$ with respect to $x$ is $28x^3$ 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: $\newline$ $y' = \ln(4)\cdot4^{7x^4} \cdot 28x^3$
Simplify Final Answer: Simplify the expression to get the final answer: $\newline$ $y' = 28\ln(4)\cdot4^{7x^4}\cdot x^3$

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