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

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Convert between exponential and logarithmic form

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Q. Find the derivative of the following function.

\newline

y=\log _{5}\left(6 x^{6}-2 x^{5}\right)

\newline

Answer:

y^{\prime}=

Identify $u$ as argument: We need to find the derivative of the function $y$ with respect to $x$ , where $y=\log_{5}(6x^{6}-2x^{5})$ . To do this, we will use the chain rule and the logarithmic differentiation rule which states that the derivative of $\log_b(u)$ with respect to $x$ is $(1/u) \cdot (du/dx) \cdot (1/\log(b))$ , where $u$ is a function of $x$ and $b$ is the base of the logarithm.
Find derivative of extit{u}: First, let's identify extit{u} as the argument of the logarithm: extit{u} = $6$ x^{ $6$ } - $2$ x^{ $5$ }. We will need to find its derivative rac{du}{dx}.
Apply chain rule: Now, let's find the derivative of $u$ with respect to $x$ . Using the power rule, we get: $\frac{du}{dx} = \frac{d}{dx}(6x^{6}) - \frac{d}{dx}(2x^{5})$ $= 6 \cdot 6x^{5} - 2 \cdot 5x^{4}$ $= 36x^{5} - 10x^{4}.$
Simplify expression: Next, we apply the chain rule and logarithmic differentiation rule to find the derivative of $y$ : $y' = \left(\frac{1}{u}\right) \cdot \left(\frac{du}{dx}\right) \cdot \left(\frac{1}{\log(5)}\right)$ $= \left(\frac{1}{6x^{6} - 2x^{5}}\right) \cdot \left(36x^{5} - 10x^{4}\right) \cdot \left(\frac{1}{\log(5)}\right).$
Check for further simplification: We can simplify the expression by multiplying the terms together: $\newline$ $y' = \frac{36x^{5} - 10x^{4}}{(6x^{6} - 2x^{5}) \cdot \log(5)}$ .
Check for further simplification: We can simplify the expression by multiplying the terms together: $\newline$ $y' = \frac{36x^{5} - 10x^{4}}{(6x^{6} - 2x^{5}) \cdot \log(5)}$ .The final step is to check if we can simplify the expression further. However, in this case, the expression is already in its simplest form, so we have our final answer.