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

Identify Function and Base: Identify the function and the base of the logarithm. We are given the function y = log_6(6x^6), where the base of the logarithm is 6.Apply Change of Base Formula: Apply the change of base formula to convert the logarithm to a natural logarithm. The change of base formula is log_b(a) = ln(a) / ln(b). Using this, we can rewrite the function as y = ln(6x^6) / ln(6).Differentiate Using Chain Rule: Differentiate the function using the chain rule. The derivative of ln(u) with respect to x is (1/u) * (du/dx). Here, u = 6x^6. So, we need to find the derivative of u with respect to x, which is du/dx = d/dx(6x^6).Calculate Derivative of u: Calculate the derivative of u = 6x^6 with respect to x. Using the power rule, the derivative of x^n with respect to x is n*x^(n-1). Therefore, du/dx = 6 * 6 * x^(6-1) = 36x^5.Substitute Derivative into y: Substitute the derivative of u into the derivative of y. The derivative of y is then y' = (1/(6x^6)) * (36x^5) / ln(6).Simplify Expression: Simplify the expression. We can cancel out an x^5 from the numerator and denominator, which gives us y' = (36/x) / ln(6).Further Simplify: Further simplify the expression. We can rewrite the derivative as y' = 36 / (x * ln(6)).

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Find the derivative of the following function.

y=log_(6)(6x^(6))
Answer:
y^(')=

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

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Precalculus

Quotient property of logarithms

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Q. Find the derivative of the following function.

\newline

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

\newline

Answer:

y^{\prime}=

Identify Function and Base: Identify the function and the base of the logarithm. $\newline$ We are given the function $y = \log_6(6x^6)$ , where the base of the logarithm is $6$ .
Apply Change of Base Formula: Apply the change of base formula to convert the logarithm to a natural logarithm. $\newline$ The change of base formula is $\log_b(a) = \frac{\ln(a)}{\ln(b)}$ . Using this, we can rewrite the function as $y = \frac{\ln(6x^6)}{\ln(6)}$ .
Differentiate Using Chain Rule: Differentiate the function using the chain rule. $\newline$ The derivative of $\ln(u)$ with respect to $x$ is $\frac{1}{u} \cdot \frac{du}{dx}$ . Here, $u = 6x^6$ . So, we need to find the derivative of $u$ with respect to $x$ , which is $\frac{du}{dx} = \frac{d}{dx}(6x^6)$ .
Calculate Derivative of u: Calculate the derivative of $u = 6x^6$ with respect to $x$ . Using the power rule, the derivative of $x^n$ with respect to $x$ is $n \cdot x^{(n-1)}$ . Therefore, $\frac{du}{dx} = 6 \cdot 6 \cdot x^{(6-1)} = 36x^5$ .
Substitute Derivative into y: Substitute the derivative of $u$ into the derivative of $y$ . The derivative of $y$ is then $y' = \frac{1}{6x^6} \cdot \frac{36x^5}{\ln(6)}$ .
Simplify Expression: Simplify the expression. $\newline$ We can cancel out an $x^5$ from the numerator and denominator, which gives us $y' = \frac{36}{x} / \ln(6)$ .
Further Simplify: Further simplify the expression. $\newline$ We can rewrite the derivative as $y' = \frac{36}{x \cdot \ln(6)}$ .