Calculus Examples

$\arctan (x^{2} + 1)$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \arctan (x)$ and $g (x) = x^{2} + 1$ .

Tap for more steps...

Step 1.1

To apply the Chain Rule, set $u$ as $x^{2} + 1$ .

$\frac{d}{d u} [\arctan (u)] \frac{d}{d x} [x^{2} + 1]$

Step 1.2

The derivative of $\arctan (u)$ with respect to $u$ is $\frac{1}{1 + u^{2}}$ .

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

Step 1.3

Replace all occurrences of $u$ with $x^{2} + 1$ .

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

Step 2

Differentiate.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$ .

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

Step 2.2

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

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

Step 2.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{1}{1 + {(x^{2} + 1)}^{2}} (2 x + 0)$

Step 2.4

Combine fractions.

Tap for more steps...

Step 2.4.1

Add $2 x$ and $0$ .

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

Step 2.4.2

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

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

Step 2.4.3

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

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

Step 3

Simplify.

Tap for more steps...

Step 3.1

Reorder terms.

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

Step 3.2

Simplify the denominator.

Tap for more steps...

Step 3.2.1

Rewrite ${(x^{2} + 1)}^{2}$ as $(x^{2} + 1) (x^{2} + 1)$ .

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

Step 3.2.2

Expand $(x^{2} + 1) (x^{2} + 1)$ using the FOIL Method.

Tap for more steps...

Step 3.2.2.1

Apply the distributive property.

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

Step 3.2.2.2

Apply the distributive property.

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

Step 3.2.2.3

Apply the distributive property.

$\frac{2 x}{x^{2} x^{2} + x^{2} \cdot 1 + 1 x^{2} + 1 \cdot 1 + 1}$

Step 3.2.3

Simplify and combine like terms.

Tap for more steps...

Step 3.2.3.1

Simplify each term.

Tap for more steps...

Step 3.2.3.1.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 3.2.3.1.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 x}{x^{2 + 2} + x^{2} \cdot 1 + 1 x^{2} + 1 \cdot 1 + 1}$

Step 3.2.3.1.1.2

Add $2$ and $2$ .

$\frac{2 x}{x^{4} + x^{2} \cdot 1 + 1 x^{2} + 1 \cdot 1 + 1}$

Step 3.2.3.1.2

Multiply $x^{2}$ by $1$ .

$\frac{2 x}{x^{4} + x^{2} + 1 x^{2} + 1 \cdot 1 + 1}$

Step 3.2.3.1.3

Multiply $x^{2}$ by $1$ .

$\frac{2 x}{x^{4} + x^{2} + x^{2} + 1 \cdot 1 + 1}$

Step 3.2.3.1.4

Multiply $1$ by $1$ .

$\frac{2 x}{x^{4} + x^{2} + x^{2} + 1 + 1}$

Step 3.2.3.2

Add $x^{2}$ and $x^{2}$ .

$\frac{2 x}{x^{4} + 2 x^{2} + 1 + 1}$

Step 3.2.4

Add $1$ and $1$ .

$\frac{2 x}{x^{4} + 2 x^{2} + 2}$