# Derivatives of logarithmic functions Calculate the derivative of the following f

Derivatives of logarithmic functions Calculate the derivative of the following functions
$y={\mathrm{log}}_{8}|\mathrm{tan}x|$
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Step 1
Given function is $y={\mathrm{log}}_{8}|\mathrm{tan}x|$.
From Property of logarithm, we have
${\mathrm{log}}_{a}b=\frac{\mathrm{ln}b}{\mathrm{ln}a}$
Therefore, given function can be written as:
$y={\mathrm{log}}_{8}|\mathrm{tan}x|$
$y=\frac{\mathrm{ln}|\mathrm{tan}x|}{\mathrm{ln}8}$
Step 2
We know that, the derivative of $\mathrm{tan}x={\mathrm{sec}}^{2}x$.
Differentiating the given function with respect to x,
$\frac{dy}{dx}=\frac{d}{dx}\left[\frac{\mathrm{ln}|\mathrm{tan}x|}{\mathrm{ln}8}\right]$
$=\frac{1}{\mathrm{ln}8}\left[\frac{1}{\mathrm{tan}x}\cdot {\mathrm{sec}}^{2}x\right]$
$=\frac{1}{\mathrm{ln}8}\left[\frac{{\mathrm{sec}}^{2}x}{\mathrm{tan}x}\right]$
Hence, the derivative of the given function is $\frac{1}{\mathrm{ln}8}\left[\frac{{\mathrm{sec}}^{2}x}{\mathrm{tan}x}\right]$