Calculus Examples

$x \arctan (x)$

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 $f (x) = x$ and $g (x) = \arctan (x)$ .

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

Step 2

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

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

Step 3

Differentiate using the Power Rule.

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Multiply $\arctan (x)$ by $1$ .

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

Step 4

To write $\arctan (x)$ as a fraction with a common denominator, multiply by $\frac{1 + x^{2}}{1 + x^{2}}$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify.

Tap for more steps...

Step 6.1

Apply the distributive property.

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

Step 6.2

Simplify the numerator.

Tap for more steps...

Step 6.2.1

Multiply $\arctan (x)$ by $1$ .

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

Step 6.2.2

Reorder factors in $x + \arctan (x) + \arctan (x) x^{2}$ .

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

Step 6.3

Reorder terms.

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