Derivatives of logarithmic functions Calculate the derivative of the following functions

$y={\mathrm{log}}_{8}\left|\mathrm{tan}x\right|$

ossidianaZ
2021-10-04
Answered

Derivatives of logarithmic functions Calculate the derivative of the following functions

$y={\mathrm{log}}_{8}\left|\mathrm{tan}x\right|$

You can still ask an expert for help

Anonym

Answered 2021-10-05
Author has **108** answers

Step 1

Given function is$y={\mathrm{log}}_{8}\left|\mathrm{tan}x\right|$ .

From Property of logarithm, we have

$\mathrm{log}}_{a}b=\frac{\mathrm{ln}b}{\mathrm{ln}a$

Therefore, given function can be written as:

$y={\mathrm{log}}_{8}\left|\mathrm{tan}x\right|$

$y=\frac{\mathrm{ln}\left|\mathrm{tan}x\right|}{\mathrm{ln}8}$

Step 2

We know that, the derivative of$\mathrm{tan}x={\mathrm{sec}}^{2}x$ .

Differentiating the given function with respect to x,

$\frac{dy}{dx}=\frac{d}{dx}\left[\frac{\mathrm{ln}\left|\mathrm{tan}x\right|}{\mathrm{ln}8}\right]$

$=\frac{1}{\mathrm{ln}8}[\frac{1}{\mathrm{tan}x}\cdot {\mathrm{sec}}^{2}x]$

$=\frac{1}{\mathrm{ln}8}\left[\frac{{\mathrm{sec}}^{2}x}{\mathrm{tan}x}\right]$

Hence, the derivative of the given function is$\frac{1}{\mathrm{ln}8}\left[\frac{{\mathrm{sec}}^{2}x}{\mathrm{tan}x}\right]$

Given function is

From Property of logarithm, we have

Therefore, given function can be written as:

Step 2

We know that, the derivative of

Differentiating the given function with respect to x,

Hence, the derivative of the given function is

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