Calculus Examples

$\arctan (- \frac{x}{2})$

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 $g (x) = - \frac{x}{2}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $- \frac{x}{2}$ .

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $- \frac{x}{2}$ .

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

Step 2

Differentiate.

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Step 2.1

Factor $- 1$ out of $- \frac{x}{2}$ .

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

Step 2.2

Simplify the expression.

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Step 2.2.1

Apply the product rule to $- (\frac{x}{2})$ .

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

Step 2.2.2

Raise $- 1$ to the power of $2$ .

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

Step 2.2.3

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

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

Step 2.3

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

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

Step 2.4

Combine fractions.

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Step 2.4.1

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

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

Step 2.4.2

Move $2$ to the left of $1 + {(\frac{x}{2})}^{2}$ .

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

Step 2.5

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

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

Step 2.6

Multiply $- 1$ by $1$ .

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

Step 3

Simplify.

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Step 3.1

Apply the product rule to $\frac{x}{2}$ .

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Combine terms.

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Step 3.3.1

Multiply $2$ by $1$ .

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

Cancel the common factor of $2$ and $4$ .

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Step 3.3.4.1

Factor $2$ out of $2 x^{2}$ .

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

Step 3.3.4.2

Cancel the common factors.

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Step 3.3.4.2.1

Factor $2$ out of $4$ .

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

Step 3.3.4.2.2

Cancel the common factor.

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

Step 3.3.4.2.3

Rewrite the expression.

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

Step 3.4

Simplify the denominator.

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

Combine the numerators over the common denominator.

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

Step 3.4.4

Multiply $2$ by $2$ .

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

Step 3.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.6

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

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