Calculus Examples

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

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 $f (x) = x^{2} + 1$ and $g (x) = \arctan (x)$ .

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

Step 2

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

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

Step 3

Differentiate.

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Add $2 x$ and $0$ .

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

Step 4

Simplify.

Tap for more steps...

Step 4.1

Reorder terms.

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

Step 4.2

Simplify each term.

Tap for more steps...

Step 4.2.1

Multiply $x^{2} + 1$ by $\frac{1}{1 + x^{2}}$ .

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

Step 4.2.2

Cancel the common factor of $x^{2} + 1$ and $1 + x^{2}$ .

Tap for more steps...

Step 4.2.2.1

Reorder terms.

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

Step 4.2.2.2

Cancel the common factor.

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

Step 4.2.2.3

Rewrite the expression.

$1 + 2 x \arctan (x)$