Find the derivative of the following function. y=ln(-x^(5)) Answer: y^(')=

Identify Function & Operation: Identify the function and the operation needed to find the derivative. 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. 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. The derivative of ln(u) with respect to u is 1/u. So, we have 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*x^(n-1).Multiply Derivatives: Multiply the derivatives from Step 3 and Step 4. We multiply 1/(-x^(5)) by -5x^(4) to get the derivative of the composite function. y' = (1/(-x^(5))) * (-5x^(4))Simplify Expression: Simplify the expression. When we multiply the two terms, we get: y' = -5x^(4) / (-x^(5)) This simplifies to: y' = 5 / x

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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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Calculus

Find derivatives using logarithmic differentiation

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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: $\newline$ $y' = \frac{5}{x}$