Calculus Examples

$8^{x} \log_{8} (x)$

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 $f (x) = 8^{x}$ and $g (x) = \log_{8} (x)$ .

$8^{x} \frac{d}{d x} [\log_{8} (x)] + \log_{8} (x) \frac{d}{d x} [8^{x}]$

The derivative of $\log_{8} (x)$ with respect to $x$ is $\frac{1}{x \ln (8)}$ .

$8^{x} \frac{1}{x \ln (8)} + \log_{8} (x) \frac{d}{d x} [8^{x}]$

Combine $8^{x}$ and $\frac{1}{x \ln (8)}$ .

$\frac{8^{x}}{x \ln (8)} + \log_{8} (x) \frac{d}{d x} [8^{x}]$

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $8$ .

$\frac{8^{x}}{x \ln (8)} + \log_{8} (x) (8^{x} \ln (8))$

Reorder terms.

$8^{x} \ln (8) \log_{8} (x) + \frac{8^{x}}{x \ln (8)}$