Find the derivative of the following function. y=log_(5)(6x^(2)+2x) Answer: y^(')=

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Find the derivative of the following function.  y=log_(5)(6x^(2)+2x) Answer:  y^(')=

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Identify Function and 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(6x^2 + 2x) with respect to x. This is a logarithmic differentiation problem where the base of the logarithm is 5.Apply Chain Rule: Apply the chain rule for differentiation. 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 = 6x^2 + 2x, the derivative with respect to x is (1/u) * (du/dx).Differentiate Inner Function: Differentiate the inner function u = 6x^2 + 2x with respect to x. The derivative of u with respect to x is du/dx = d/dx(6x^2) + d/dx(2x) = 12x + 2.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 use this to convert the base 5 logarithm to a natural logarithm: log_5(u) = ln(u) / ln(5).Combine Results for Derivative: Combine the results from Steps 2, 3, and 4 to find the derivative of y. Using the chain rule and the change of base formula, the derivative of y with respect to x is y' = (1/(6x^2 + 2x)) * (12x + 2) / ln(5).Simplify Derivative Expression: Simplify the expression for the derivative. y' = (12x + 2) / ((6x^2 + 2x) * ln(5)). This is the simplified form of the derivative.

Find the derivative of the following function. $\newline$ $y=\log _{5}\left(6 x^{2}+2 x\right)$ $\newline$ Answer: $y^{\prime}=$

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Find the derivative of the following function. $\newline$ $y=\log _{5}\left(6 x^{2}+2 x\right)$ $\newline$ Answer: $y^{\prime}=$