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Find the derivative of the following function.

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

Find the derivative of the following function. $\newline$ $y=\ln \left(x^{4}\right)$ $\newline$ Answer: $y^{\prime}=$

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Find derivatives using logarithmic differentiation

Full solution

Q. Find the derivative of the following function.

\newline

y=\ln \left(x^{4}\right)

\newline

Answer:

y^{\prime}=

Identify function: Identify the function to differentiate. $\newline$ We are given the function $y = \ln(x^{4})$ . We need to find its derivative with respect to $x$ .
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 $\ln(u)$ and the inner function is $u = x^{4}$ .
Differentiate outer function: Differentiate the outer function with respect to the inner function. $\newline$ The derivative of $\ln(u)$ with respect to $u$ is $\frac{1}{u}$ . So, we have $\frac{1}{x^{4}}$ for the outer function's derivative.
Differentiate inner function: Differentiate the inner function with respect to $x$ . The derivative of $x^{4}$ with respect to $x$ is $4x^{3}$ .
Multiply derivatives: Multiply the derivatives from Step $3$ and Step $4$ . $\newline$ We multiply $\frac{1}{x^{4}}$ by $4x^{3}$ to get the derivative of the composite function.
Simplify expression: Simplify the expression. $\newline$ Multiplying $\frac{1}{x^{4}}$ by $4x^{3}$ gives us $\frac{4x^{3}}{x^{4}}$ . This simplifies to $\frac{4}{x}$ .
Write final answer: Write the final answer. $\newline$ The derivative of $y = \ln(x^{4})$ with respect to $x$ is $y' = \frac{4}{x}$ .

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