Calculus Examples

$f (x) = e^{x} \log_{4} (x)$

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = e^{x}$ and $g (x) = \log_{4} (x)$ .

$e^{x} \frac{d}{d x} [\log_{4} (x)] + \log_{4} (x) \frac{d}{d x} [e^{x}]$

The derivative of $\log_{4} (x)$ with respect to $x$ is $\frac{1}{x \ln (4)}$ .

$e^{x} \frac{1}{x \ln (4)} + \log_{4} (x) \frac{d}{d x} [e^{x}]$

Combine $e^{x}$ and $\frac{1}{x \ln (4)}$ .

$\frac{e^{x}}{x \ln (4)} + \log_{4} (x) \frac{d}{d x} [e^{x}]$

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$\frac{e^{x}}{x \ln (4)} + \log_{4} (x) e^{x}$

Reorder terms.

$e^{x} \log_{4} (x) + \frac{e^{x}}{x \ln (4)}$