Calculus Examples

$f (x) = (1 + 4 x^{2}) \arctan (2 x)$

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 $f (x) = 1 + 4 x^{2}$ and $g (x) = \arctan (2 x)$ .

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

Step 2

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 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Differentiate.

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Step 3.1

Factor $2$ out of $2 x$ .

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

Step 3.2

Simplify the expression.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

By the Sum Rule, the derivative of $1 + 4 x^{2}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [4 x^{2}]$ .

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

Step 3.8

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$(1 + 4 x^{2}) \frac{2}{1 + 4 x^{2}} + \arctan (2 x) (0 + \frac{d}{d x} [4 x^{2}])$

Step 3.9

Add $0$ and $\frac{d}{d x} [4 x^{2}]$ .

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

Step 3.10

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

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

Step 3.11

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$(1 + 4 x^{2}) \frac{2}{1 + 4 x^{2}} + \arctan (2 x) (4 (2 x))$

Step 3.12

Multiply $2$ by $4$ .

$(1 + 4 x^{2}) \frac{2}{1 + 4 x^{2}} + \arctan (2 x) (8 x)$

Step 4

Simplify.

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Step 4.1

Reorder terms.

$(1 + 4 x^{2}) \frac{2}{1 + 4 x^{2}} + 8 x \arctan (2 x)$

Step 4.2

Cancel the common factor of $1 + 4 x^{2}$ .

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Step 4.2.1

Cancel the common factor.

$(1 + 4 x^{2}) \frac{2}{1 + 4 x^{2}} + 8 x \arctan (2 x)$

Step 4.2.2

Rewrite the expression.

$2 + 8 x \arctan (2 x)$