Calculus Examples

$y = \ln (x^{2})$ , $x = - 5$

Step 1

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 $g (x) = x^{2}$ .

Tap for more steps...

Step 1.1

To apply the Chain Rule, set $u$ as $x^{2}$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x^{2}]$

Step 1.2

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

$\frac{1}{u} \frac{d}{d x} [x^{2}]$

Step 1.3

Replace all occurrences of $u$ with $x^{2}$ .

$\frac{1}{x^{2}} \frac{d}{d x} [x^{2}]$

Step 2

Differentiate using the Power Rule.

Tap for more steps...

Step 2.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{1}{x^{2}} (2 x)$

Step 2.2

Simplify terms.

Tap for more steps...

Step 2.2.1

Combine $2$ and $\frac{1}{x^{2}}$ .

$\frac{2}{x^{2}} x$

Step 2.2.2

Combine $\frac{2}{x^{2}}$ and $x$ .

$\frac{2 x}{x^{2}}$

Step 2.2.3

Cancel the common factor of $x$ and $x^{2}$ .

Tap for more steps...

Step 2.2.3.1

Factor $x$ out of $2 x$ .

$\frac{x \cdot 2}{x^{2}}$

Step 2.2.3.2

Cancel the common factors.

Tap for more steps...

Step 2.2.3.2.1

Factor $x$ out of $x^{2}$ .

$\frac{x \cdot 2}{x \cdot x}$

Step 2.2.3.2.2

Cancel the common factor.

$\frac{x \cdot 2}{x \cdot x}$

Step 2.2.3.2.3

Rewrite the expression.

$\frac{2}{x}$

Step 3

Evaluate the derivative at $x = - 5$ .

$\frac{2}{- 5}$

Step 4

Move the negative in front of the fraction.

$- \frac{2}{5}$