Calculus Examples

$y = \arctan (3 x)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\arctan (3 x))$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

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 $g (x) = 3 x$ .

Tap for more steps...

Step 3.1.1

To apply the Chain Rule, set $u$ as $3 x$ .

$\frac{d}{d u} [\arctan (u)] \frac{d}{d x} [3 x]$

Step 3.1.2

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

$\frac{1}{1 + u^{2}} \frac{d}{d x} [3 x]$

Step 3.1.3

Replace all occurrences of $u$ with $3 x$ .

$\frac{1}{1 + {(3 x)}^{2}} \frac{d}{d x} [3 x]$

Step 3.2

Differentiate.

Tap for more steps...

Step 3.2.1

Factor $3$ out of $3 x$ .

$\frac{1}{1 + {(3 (x))}^{2}} \frac{d}{d x} [3 x]$

Step 3.2.2

Simplify the expression.

Tap for more steps...

Step 3.2.2.1

Apply the product rule to $3 (x)$ .

$\frac{1}{1 + 3^{2} x^{2}} \frac{d}{d x} [3 x]$

Step 3.2.2.2

Raise $3$ to the power of $2$ .

$\frac{1}{1 + 9 x^{2}} \frac{d}{d x} [3 x]$

Step 3.2.3

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$\frac{1}{1 + 9 x^{2}} (3 \frac{d}{d x} [x])$

Step 3.2.4

Combine $3$ and $\frac{1}{1 + 9 x^{2}}$ .

$\frac{3}{1 + 9 x^{2}} \frac{d}{d x} [x]$

Step 3.2.5

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

$\frac{3}{1 + 9 x^{2}} \cdot 1$

Step 3.2.6

Simplify the expression.

Tap for more steps...

Step 3.2.6.1

Multiply $\frac{3}{1 + 9 x^{2}}$ by $1$ .

$\frac{3}{1 + 9 x^{2}}$

Step 3.2.6.2

Reorder terms.

$\frac{3}{9 x^{2} + 1}$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = \frac{3}{9 x^{2} + 1}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{3}{9 x^{2} + 1}$