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Find the derivative of the following function.

y=log_(6)(-x^(5))
Answer:
y^(')=

Find the derivative of the following function. $\newline$ $y=\log _{6}\left(-x^{5}\right)$ $\newline$ Answer: $y^{\prime}=$

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Find derivatives using logarithmic differentiation

Full solution

Q. Find the derivative of the following function.

\newline

y=\log _{6}\left(-x^{5}\right)

\newline

Answer:

y^{\prime}=

Identify Function and Derivative: Identify the function and the type of derivative to be found. $\newline$ We are given the function $y=\log_6(-x^5)$ and we need to find its derivative with respect to $x$ .
Apply Chain Rule: Apply the chain rule for derivatives. 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 $\log_6(u)$ and the inner function is $u=-x^5$ .
Find Derivative of Outer Function: Find the derivative of the outer function. $\newline$ The derivative of $\log_6(u)$ with respect to $u$ is $\frac{1}{u\ln(6)}$ because the derivative of $\log_b(u)$ is $\frac{1}{u\ln(b)}$ .
Find Derivative of Inner Function: Find the derivative of the inner function. $\newline$ The derivative of $-x^5$ with respect to $x$ is $-5x^4$ .
Apply Chain Rule with Derivatives: Apply the chain rule using the derivatives from steps $3$ and $4$ . The derivative of $y$ with respect to $x$ is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. So, $y' = \frac{1}{(-x^5\ln(6))}\cdot(-5x^4)$ .
Simplify Expression: Simplify the expression. $\newline$ $y' = \frac{-5x^4}{-x^5\ln(6)}$ . $\newline$ We can simplify this by canceling out an $x^4$ from the numerator and denominator. $\newline$ $y' = \frac{-5}{-x\ln(6)}$ .
Cancel Negative Signs: Simplify further by canceling out the negative signs. $\newline$ $y' = \frac{5}{x\ln(6)}$ .

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