Find the derivative of the following function. y=log_(8)(-9x^(5)) Answer: y^(')=

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Find the derivative of the following function.  y=log_(8)(-9x^(5)) Answer:  y^(')=

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Understand and apply chain rule: Understand the function and apply the chain rule. The function y=log_(8)(-9x^(5)) is a logarithmic function with base 8. 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.Convert base to natural logarithm: Convert the base of the logarithm to the natural logarithm (ln). The derivative of a logarithm with base 8 can be found by converting it to a natural logarithm using the change of base formula: log_(a)(b) = ln(b)/ln(a). So, y=log_(8)(-9x^(5)) becomes y=ln(-9x^(5))/ln(8).Differentiate outer function: Differentiate the outer function. The outer function is ln(-9x^(5)). The derivative of ln(u), where u is a function of x, is 1/u * u', where u' is the derivative of u. So, the derivative of ln(-9x^(5)) with respect to x is 1/(-9x^(5)) times the derivative of -9x^(5) with respect to x.Differentiate inner function: Differentiate the inner function. The inner function is -9x^(5). The derivative of -9x^(5) with respect to x is -45x^(4) (using the power rule: d/dx [x^n] = nx^(n-1)).Apply chain rule: Apply the chain rule. Now, we multiply the derivative of the outer function by the derivative of the inner function. This gives us (1/(-9x^(5))) * (-45x^(4)).Simplify expression: Simplify the expression. Simplifying the expression (1/(-9x^(5))) * (-45x^(4)) gives us -45x^(4)/(-9x^(5)) which simplifies to 5/x.Include base conversion derivative: Include the derivative of the base conversion. We must not forget to include the denominator ln(8) from the base conversion in Step 2. The final derivative of the function y with respect to x is (5/x)/ln(8).Write final answer: Write the final answer. The final answer for the derivative of the function y=log_(8)(-9x^(5)) is y' = (5/x)/ln(8).

Find the derivative of the following function. $\newline$ $y=\log _{8}\left(-9 x^{5}\right)$ $\newline$ Answer: $y^{\prime}=$

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Find the derivative of the following function.\newliney=log⁡8(−9x5) y=\log _{8}\left(-9 x^{5}\right) y=log8​(−9x5)\newlineAnswer: y′= y^{\prime}= y′=

Find the derivative of the following function. $\newline$ $y=\log _{8}\left(-9 x^{5}\right)$ $\newline$ Answer: $y^{\prime}=$