Calculus Examples

$\frac{d}{d x} (\arctan (x) + 2 \sqrt{x})$

Step 1

By the Sum Rule, the derivative of $\arctan (x) + 2 \sqrt{x}$ with respect to $x$ is $\frac{d}{d x} [\arctan (x)] + \frac{d}{d x} [2 \sqrt{x}]$ .

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

Step 2

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

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

Step 3

Evaluate $\frac{d}{d x} [2 \sqrt{x}]$ .

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 3.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{\frac{1}{2}}$ with respect to $x$ is $2 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

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

Step 3.3

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

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

Step 3.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3.5

Combine $- 1$ and $\frac{2}{2}$ .

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

Step 3.6

Combine the numerators over the common denominator.

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

Step 3.7

Simplify the numerator.

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Step 3.7.1

Multiply $- 1$ by $2$ .

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

Step 3.7.2

Subtract $2$ from $1$ .

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

Step 3.8

Move the negative in front of the fraction.

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

Step 3.9

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

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

Step 3.10

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

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

Step 3.11

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.12

Cancel the common factor.

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

Step 3.13

Rewrite the expression.

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

Step 4

Simplify.

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Step 4.1

Combine terms.

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

Write each expression with a common denominator of $(1 + x^{2}) x^{\frac{1}{2}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.3.1

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

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

Step 4.1.3.2

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

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

Step 4.1.3.3

Reorder the factors of $(1 + x^{2}) x^{\frac{1}{2}}$ .

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

Step 4.1.4

Combine the numerators over the common denominator.

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

Step 4.2

Reorder terms.

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