Find the derivative of the following function. y=ln(x^(4)) Answer: y^(')=

Identify function: Identify the function to differentiate. 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. 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. The derivative of ln(u) with respect to u is 1/u. So, we have 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. We multiply 1/(x^(4)) by 4x^(3) to get the derivative of the composite function.Simplify expression: Simplify the expression. Multiplying 1/(x^(4)) by 4x^(3) gives us (4x^(3))/(x^(4)). This simplifies to 4/x.Write final answer: Write the final answer. The derivative of y = ln(x^(4)) with respect to x is y' = 4/x.

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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

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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 $y' = \frac{4}{x}$ .