Find the derivative of the following function. y=log_(9)(-6x^(6)-6x^(5)) Answer: y^(')=

Apply Chain Rule: First, we need to apply the chain rule to differentiate the logarithmic function with base 9. 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. The derivative of log_9(u) with respect to u is 1/(u * ln(9)), where u is the inner function. In this case, u = -6x^6 - 6x^5.Find Derivative of Inner Function: Next, we need to find the derivative of the inner function u with respect to x, which is -6x^6 - 6x^5. Using the power rule, the derivative of -6x^6 is -36x^5, and the derivative of -6x^5 is -30x^4. So, the derivative of u with respect to x is du/dx = -36x^5 - 30x^4.Combine Results: Now, we can combine the results from the first two steps to find the derivative of y with respect to x. Using the chain rule, we get: y' = (1/(u * ln(9))) * du/dx Substituting u = -6x^6 - 6x^5 and du/dx = -36x^5 - 30x^4, we have: y' = (1/((-6x^6 - 6x^5) * ln(9))) * (-36x^5 - 30x^4)Simplify Expression: Finally, we simplify the expression for y' by distributing the derivative of the inner function across the numerator. y' = (-36x^5 - 30x^4) / ((-6x^6 - 6x^5) * ln(9)) This is the derivative of the function y with respect to x.

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Find the derivative of the following function.

y=log_(9)(-6x^(6)-6x^(5))
Answer:
y^(')=

Find the derivative of the following function. $\newline$ $y=\log _{9}\left(-6 x^{6}-6 x^{5}\right)$ $\newline$ Answer: $y^{\prime}=$

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Math Problems

Precalculus

Quotient property of logarithms

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Q. Find the derivative of the following function.

\newline

y=\log _{9}\left(-6 x^{6}-6 x^{5}\right)

\newline

Answer:

y^{\prime}=

Apply Chain Rule: First, we need to apply the chain rule to differentiate the logarithmic function with base $9$ . 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. $\newline$ The derivative of $\log_9(u)$ with respect to $u$ is $1/(u \cdot \ln(9))$ , where $u$ is the inner function. $\newline$ In this case, $u = -6x^6 - 6x^5$ .
Find Derivative of Inner Function: Next, we need to find the derivative of the inner function $u$ with respect to $x$ , which is $-6x^6 - 6x^5$ . Using the power rule, the derivative of $-6x^6$ is $-36x^5$ , and the derivative of $-6x^5$ is $-30x^4$ . So, the derivative of $u$ with respect to $x$ is $\frac{du}{dx} = -36x^5 - 30x^4$ .
Combine Results: Now, we can combine the results from the first two steps to find the derivative of $y$ with respect to $x$ . Using the chain rule, we get: $y' = \frac{1}{(u \cdot \ln(9))} \cdot \frac{du}{dx}$ Substituting $u = -6x^6 - 6x^5$ and $\frac{du}{dx} = -36x^5 - 30x^4$ , we have: $y' = \frac{1}{((-6x^6 - 6x^5) \cdot \ln(9))} \cdot (-36x^5 - 30x^4)$
Simplify Expression: Finally, we simplify the expression for $y'$ by distributing the derivative of the inner function across the numerator. $\newline$ $y' = \frac{-36x^5 - 30x^4}{(-6x^6 - 6x^5) \cdot \ln(9)}$ $\newline$ This is the derivative of the function $y$ with respect to $x$ .