Calculus Examples

$9^{x} \log_{5} (x)$

Step 1

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 $f (x) = 9^{x}$ and $g (x) = \log_{5} (x)$ .

$9^{x} \frac{d}{d x} [\log_{5} (x)] + \log_{5} (x) \frac{d}{d x} [9^{x}]$

Step 2

The derivative of $\log_{5} (x)$ with respect to $x$ is $\frac{1}{x \ln (5)}$ .

$9^{x} \frac{1}{x \ln (5)} + \log_{5} (x) \frac{d}{d x} [9^{x}]$

Step 3

Combine $9^{x}$ and $\frac{1}{x \ln (5)}$ .

$\frac{9^{x}}{x \ln (5)} + \log_{5} (x) \frac{d}{d x} [9^{x}]$

Step 4

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $9$ .

$\frac{9^{x}}{x \ln (5)} + \log_{5} (x) (9^{x} \ln (9))$

Step 5

Reorder terms.

$9^{x} \ln (9) \log_{5} (x) + \frac{9^{x}}{x \ln (5)}$