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Find the derivative of the following function.

y=ln(-x^(5))
Answer:
y^(')=

Find the derivative of the following function. $\newline$ $y=\ln \left(-x^{5}\right)$ $\newline$ 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^{5}\right)

\newline

Answer:

y^{\prime}=

Identify Function & Operation: Identify the function and the operation needed to find the derivative. $\newline$ We have the function $y = \ln(-x^{5})$ . To find the derivative $y'$ , we will use the chain rule because the function is a composition of two functions: the natural logarithm function and the function $-x^{5}$ .
Apply Chain Rule: Apply the chain rule. $\newline$ 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 $u = -x^{5}$ .
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 $\frac{1}{-x^{5}}$ for the derivative of the outer function.
Differentiate Inner Function: Differentiate the inner function with respect to $x$ . The derivative of $-x^{5}$ with respect to $x$ is $-5x^{4}$ because we apply the power rule which states that the derivative of $x^n$ is $n\cdot x^{(n-1)}$ .
Multiply Derivatives: Multiply the derivatives from Step $3$ and Step $4$ . $\newline$ We multiply $\frac{1}{(-x^{5})}$ by $-5x^{4}$ to get the derivative of the composite function. $\newline$ $y' = \left(\frac{1}{(-x^{5})}\right) * (-5x^{4})$
Simplify Expression: Simplify the expression. $\newline$ When we multiply the two terms, we get: $\newline$ $y' = \frac{-5x^{4}}{-x^{5}}$ $\newline$ This simplifies to: $\newline$ $y' = \frac{5}{x}$

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