# Math Help - Derivative

1. ## Derivative

Differentiate log(base 7)(4x^2+1)

2. Originally Posted by Asuhuman18
log(base 7)(4x^2+1)
$\frac{d}{dx} \log_b(u) = \frac{1}{\ln{b}} \cdot \frac{u'}{u}$

3. Originally Posted by Asuhuman18
log(base 7)(4x^2+1)
Use the change of base rule to turn it into a natural logarithm function.

$y = \log_7{(4x^2 + 1)}$

$= \frac{\ln{(4x^2 + 1)}}{\ln{7}}$

$= \frac{1}{\ln{7}}\cdot\ln{(4x^2 + 1)}$.

Let $u = 4x^2 + 1$ so that $y= \frac{1}{\ln{7}}\cdot\ln{u}$.

$\frac{du}{dx} = 8x$

$\frac{dy}{du} = \frac{1}{\ln{7}}\cdot\frac{1}{u}$

$= \frac{1}{(4x^2 + 1)\ln{7}}$.

Therefore

$\frac{dy}{dx} = \frac{8x}{(4x^2 + 1)\ln{7}}$.