Find the derivative of the following function. y=log_(6)(-8x^(6)) Answer: y^(')=

Understand and Apply Chain Rule: Understand the function and apply the chain rule. The function y=log_(6)(-8x^(6)) is a logarithmic function with base 6. To find the derivative, we need to use the chain rule, which 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.Differentiate Outer Function: Differentiate the outer function. The outer function is log_(6)(u), where u = -8x^(6). The derivative of log_(6)(u) with respect to u is 1/(u * ln(6)) according to the logarithmic differentiation rule.Differentiate Inner Function: Differentiate the inner function. The inner function is u = -8x^(6). The derivative of -8x^(6) with respect to x is -48x^(5) by using the power rule.Apply Chain Rule: Apply the chain rule. Now we multiply the derivative of the outer function by the derivative of the inner function to get the derivative of the composite function. y^(') = (1/(-8x^(6) * ln(6))) * (-48x^(5))Simplify Expression: Simplify the expression. We can simplify the expression by multiplying the constants and combining like terms. y^(') = -48 / (-8x * ln(6)) * x^(5) y^(') = 6 / (x * ln(6))

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Find the derivative of the following function.

y=log_(6)(-8x^(6))
Answer:
y^(')=

Find the derivative of the following function. $\newline$ $y=\log _{6}\left(-8 x^{6}\right)$ $\newline$ Answer: $y^{\prime}=$

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Precalculus

Quotient property of logarithms

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Q. Find the derivative of the following function.

\newline

y=\log _{6}\left(-8 x^{6}\right)

\newline

Answer:

y^{\prime}=

Understand and Apply Chain Rule: Understand the function and apply the chain rule. $\newline$ The function $y=\log_{6}(-8x^{6})$ is a logarithmic function with base $6$ . To find the derivative, we need to use the chain rule, which 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.
Differentiate Outer Function: Differentiate the outer function. $\newline$ The outer function is $\log_{6}(u)$ , where $u = -8x^{6}$ . The derivative of $\log_{6}(u)$ with respect to $u$ is $\frac{1}{u \cdot \ln(6)}$ according to the logarithmic differentiation rule.
Differentiate Inner Function: Differentiate the inner function. $\newline$ The inner function is $u = -8x^{6}$ . The derivative of $-8x^{6}$ with respect to $x$ is $-48x^{5}$ by using the power rule.
Apply Chain Rule: Apply the chain rule. $\newline$ Now we multiply the derivative of the outer function by the derivative of the inner function to get the derivative of the composite function. $\newline$ $y' = \frac{1}{-8x^{6} \cdot \ln(6)} \cdot (-48x^{5})$
Simplify Expression: Simplify the expression. $\newline$ We can simplify the expression by multiplying the constants and combining like terms. $\newline$ $y' = \frac{-48}{-8x \cdot \ln(6)} \cdot x^{5}$ $\newline$ $y' = \frac{6}{x \cdot \ln(6)}$