Calculus Examples

$y = \frac{x}{\arctan (x)}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x$ and $g (x) = \arctan (x)$ .

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

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 $n = 1$ .

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

Step 2.2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Rewrite $\frac{\frac{\arctan (x) (1 + x^{2}) - x}{1 + x^{2}}}{{\arctan (x)}^{2}}$ as a product.

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

Step 8

Multiply $\frac{\arctan (x) (1 + x^{2}) - x}{1 + x^{2}}$ by $\frac{1}{{\arctan (x)}^{2}}$ .

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

Step 9

Simplify.

Tap for more steps...

Step 9.1

Apply the distributive property.

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

Step 9.2

Apply the distributive property.

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

Step 9.3

Simplify the numerator.

Tap for more steps...

Step 9.3.1

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

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

Step 9.3.2

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

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

Step 9.4

Multiply ${\arctan (x)}^{2}$ by $1$ .

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

Step 9.5

Reorder terms.

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