Calculus Examples

$\ln (x^{16})$

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = x^{16}$ .

Tap for more steps...

To apply the Chain Rule, set $u$ as $x^{16}$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x^{16}]$

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

$\frac{1}{u} \frac{d}{d x} [x^{16}]$

Replace all occurrences of $u$ with $x^{16}$ .

$\frac{1}{x^{16}} \frac{d}{d x} [x^{16}]$

Differentiate using the Power Rule.

Tap for more steps...

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 16$ .

$\frac{1}{x^{16}} (16 x^{15})$

Simplify terms.

Tap for more steps...

Combine $16$ and $\frac{1}{x^{16}}$ .

$\frac{16}{x^{16}} x^{15}$

Combine $\frac{16}{x^{16}}$ and $x^{15}$ .

$\frac{16 x^{15}}{x^{16}}$

Cancel the common factor of $x^{15}$ and $x^{16}$ .

Tap for more steps...

Factor $x^{15}$ out of $16 x^{15}$ .

$\frac{x^{15} \cdot 16}{x^{16}}$

Cancel the common factors.

Tap for more steps...

Factor $x^{15}$ out of $x^{16}$ .

$\frac{x^{15} \cdot 16}{x^{15} x}$

Cancel the common factor.

$\frac{x^{15} \cdot 16}{x^{15} x}$

Rewrite the expression.

$\frac{16}{x}$