# Math Help - logarithmic differentiation

1. ## logarithmic differentiation

Find the derivative of

y=(1+cosx)^(2x)

by taking the ln of both sides then using implicit differentiation.

can someone show me the steps for this?

2. Hello, thecount!

Find the derivative of: . $y \:=\:(1+\cos x)^{2x}$

by taking the $\ln$ of both sides then using implicit differentiation.

Take the $\ln$ of both sides: . $\ln(y) \:=\:\ln(1 + \cos x)^{2x} \quad\Rightarrow\quad \ln(y) \:=\:2x\cdot\ln(1 + \cos x)$

Differentiate implicitly (product rule):

. . $\frac{1}{y}\,\frac{dy}{dx} \;=\;2x\cdot\frac{1}{1+\cos x}\cdot(-\sin x) + 2\cdot\ln(1+\cos x) \;=\;\frac{-2\sin x}{1+\cos x} + 2\ln(1+\cos x)$

Multiply by $y\!:\;\;\frac{dy}{dx} \;=\;y\cdot\bigg[\frac{-2\sin x}{1 + \cos x} + 2\ln(1+\cos x)\bigg]$

Therefore: . $\frac{dy}{dx} \;=\;(1+\cos x)^{2x}\left[\frac{-2\sin x}{1 + \cos x} + 2\ln(1 + \cos x)\right]$