Calculus Examples

$f (x) = \arctan (2 x)$

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 $g (x) = 2 x$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $2 x$ .

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $2 x$ .

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

Step 2

Differentiate.

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Step 2.1

Factor $2$ out of $2 x$ .

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

Step 2.2

Simplify the expression.

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Step 2.2.1

Apply the product rule to $2 (x)$ .

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

Step 2.2.2

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

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

Step 2.3

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 2.4

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

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

Step 2.5

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

$\frac{2}{1 + 4 x^{2}} \cdot 1$

Step 2.6

Simplify the expression.

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Step 2.6.1

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

$\frac{2}{1 + 4 x^{2}}$

Step 2.6.2

Reorder terms.

$\frac{2}{4 x^{2} + 1}$