Calculus Examples

$\arctan (4 X)$

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 $g (X) = 4 X$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $4 X$ .

$\frac{d}{d u} [\arctan (u)] \frac{d}{d X} [4 X]$

Step 1.2

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

$\frac{1}{1 + u^{2}} \frac{d}{d X} [4 X]$

Step 1.3

Replace all occurrences of $u$ with $4 X$ .

$\frac{1}{1 + {(4 X)}^{2}} \frac{d}{d X} [4 X]$

Step 2

Differentiate.

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Step 2.1

Factor $4$ out of $4 X$ .

$\frac{1}{1 + {(4 (X))}^{2}} \frac{d}{d X} [4 X]$

Step 2.2

Simplify the expression.

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Step 2.2.1

Apply the product rule to $4 (X)$ .

$\frac{1}{1 + 4^{2} X^{2}} \frac{d}{d X} [4 X]$

Step 2.2.2

Raise $4$ to the power of $2$ .

$\frac{1}{1 + 16 X^{2}} \frac{d}{d X} [4 X]$

Step 2.3

Since $4$ is constant with respect to $X$ , the derivative of $4 X$ with respect to $X$ is $4 \frac{d}{d X} [X]$ .

$\frac{1}{1 + 16 X^{2}} (4 \frac{d}{d X} [X])$

Step 2.4

Combine $4$ and $\frac{1}{1 + 16 X^{2}}$ .

$\frac{4}{1 + 16 X^{2}} \frac{d}{d X} [X]$

Step 2.5

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

$\frac{4}{1 + 16 X^{2}} \cdot 1$

Step 2.6

Simplify the expression.

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Step 2.6.1

Multiply $\frac{4}{1 + 16 X^{2}}$ by $1$ .

$\frac{4}{1 + 16 X^{2}}$

Step 2.6.2

Reorder terms.

$\frac{4}{16 X^{2} + 1}$