Find the derivative of the following function. y=log_(5)(-4x^(5)) Answer: y^(')=

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Find the derivative of the following function.  y=log_(5)(-4x^(5)) Answer:  y^(')=

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Identify Function & Derivative Type: Identify the function and the type of derivative to be found. We need to find the derivative of the function y = log_5(-4x^5) with respect to x. This involves using the chain rule and the logarithmic differentiation rule.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. For the logarithmic function y = log_5(u), where u = -4x^5, the derivative with respect to x is (1/u) * (du/dx).Differentiate Inner Function: Differentiate the inner function u = -4x^5 with respect to x. The derivative of u with respect to x is du/dx = -4 * 5x^(5-1) = -20x^4.Apply Change of Base Formula: Apply the change of base formula for logarithms. The change of base formula is log_b(a) = log_c(a) / log_c(b). We can rewrite y as log_5(-4x^5) = log(-4x^5) / log(5), where the base c of the logarithm is the natural logarithm (base e).Combine Results for Derivative: Combine the results from Steps 2, 3, and 4 to find the derivative. Using the chain rule, the derivative of y with respect to x is y' = (1/(-4x^5)) * (-20x^4) / log(5).Simplify Derivative Expression: Simplify the expression for the derivative. y' = (-20x^4 / (-4x^5)) / log(5) = (20 / (4x)) / log(5) = 5 / (x * log(5)).

Find the derivative of the following function. $\newline$ $y=\log _{5}\left(-4 x^{5}\right)$ $\newline$ Answer: $y^{\prime}=$

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Find the derivative of the following function.\newliney=log⁡5(−4x5) y=\log _{5}\left(-4 x^{5}\right) y=log5​(−4x5)\newlineAnswer: y′= y^{\prime}= y′=

Find the derivative of the following function. $\newline$ $y=\log _{5}\left(-4 x^{5}\right)$ $\newline$ Answer: $y^{\prime}=$