What is the derivative of f(x)=log_4(e^x+3) ?

Aug 6, 2014

First, we will rewrite the function in terms of natural logarithms, using the change-of-base rule:

$f \left(x\right) = \ln \frac{{e}^{x} + 3}{\ln} 4$

Differentiating will require use of the chain rule:

$\frac{d}{\mathrm{dx}} f \left(x\right) = \frac{1}{\ln} 4 \cdot \frac{d}{d \left({e}^{x} + 3\right)} \left[\ln \left({e}^{x} + 3\right)\right] \cdot \frac{d}{\mathrm{dx}} \left[{e}^{x} + 3\right]$

We know that since the derivative of $\ln x$ with respect to $x$ is $\frac{1}{x}$, then the derivative of $\ln \left({e}^{x} + 3\right)$ with respect to ${e}^{x} + 3$ will be $\frac{1}{{e}^{x} + 3}$. We also know that the derivative of ${e}^{x} + 3$ with respect to $x$ will simply be ${e}^{x}$:

$\frac{d}{\mathrm{dx}} f \left(x\right) = \frac{1}{\ln} 4 \cdot \frac{1}{{e}^{x} + 3} \cdot \left({e}^{x}\right)$

Simplifying yields:

$\frac{d}{\mathrm{dx}} f \left(x\right) = \frac{{e}^{x}}{\ln 4 \left({e}^{x} + 3\right)}$