Calculus Examples

$3^{x} \log_{3} (x)$

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 $f (x) = 3^{x}$ and $g (x) = \log_{3} (x)$ .

$3^{x} \frac{d}{d x} [\log_{3} (x)] + \log_{3} (x) \frac{d}{d x} [3^{x}]$

Step 2

The derivative of $\log_{3} (x)$ with respect to $x$ is $\frac{1}{x \ln (3)}$ .

$3^{x} \frac{1}{x \ln (3)} + \log_{3} (x) \frac{d}{d x} [3^{x}]$

Step 3

Combine $3^{x}$ and $\frac{1}{x \ln (3)}$ .

$\frac{3^{x}}{x \ln (3)} + \log_{3} (x) \frac{d}{d x} [3^{x}]$

Step 4

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $3$ .

$\frac{3^{x}}{x \ln (3)} + \log_{3} (x) (3^{x} \ln (3))$

Step 5

Reorder terms.

$3^{x} \ln (3) \log_{3} (x) + \frac{3^{x}}{x \ln (3)}$