1. ## Logarithmic Differentiation

Find f'(x) using logarithmic differentiation:
f(x)=(4x+4)^(3x)
Thanks for the help.

2. Hello, nystudent2729!

Find $f'(x)$ using logarithmic differentiation: . $f(x)\:=\:(4x+4)^{3x}$
We have: . $y \;=\;(4x+4)^{3x}$

Take logs: . $\ln(y) \;=\;\ln(4x+4)^{3x}$

. . . . . . . . $\ln(y) \;=\;3x\!\cdot\!\ln(4x+4)$

Differentiate: . $\frac{1}{y}\!\cdot\!\frac{dy}{dx} \;=\;3x\!\cdot\!\frac{4}{4x+4} + 3\!\cdot\!\ln(4x+4) \;=\;3\bigg[\frac{x}{x+1} + \ln(4x+4)\bigg]$

. . Then: . $\frac{dy}{dx} \;=\;3y\bigg[\frac{x}{x+1} + \ln(4x+4)\bigg]$

Therefore: . $\frac{dy}{dx} \;=\;3(4x+4)^{3x}\left[\frac{x}{x+1} + \ln(4x+4)\right]$