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

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

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Identify Function & Rule: Identify the function and the rule to use for differentiation. We have the function y = ln(-8x^(5)). To find the derivative, we will 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 the natural logarithm ln(u), where u = -8x^(5). The derivative of ln(u) with respect to u is 1/u.Differentiate Inner Function: Differentiate the inner function. The inner function is u = -8x^(5). The derivative of -8x^(5) with respect to x is -8 * 5x^(5-1) = -40x^(4).Apply Chain Rule: Apply the chain rule. Using the chain rule, the derivative of y with respect to x is the derivative of the outer function times the derivative of the inner function. So, y' = (1/u) * (-40x^(4)).Substitute Inner Function: Substitute the inner function back into the derivative. Substitute u = -8x^(5) back into the derivative to get y' = (1/(-8x^(5))) * (-40x^(4)).Simplify Expression: Simplify the expression. Simplify the derivative to get y' = -40x^(4) / (-8x^(5)) = 40x^(4) / (8x^(5)).Cancel Common Factors: Cancel out common factors. Cancel x^(4) from the numerator and denominator, and simplify the constants to get y' = 40 / (8x) = 5 / x.Check for Errors: Check for any mathematical errors. Review the steps to ensure there are no mathematical errors in the differentiation process.

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

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

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