Calculus Examples

$\arctan (\frac{x}{3})$

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 $g (x) = \frac{x}{3}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{x}{3}$ .

$\frac{d}{d u} [\arctan (u)] \frac{d}{d x} [\frac{x}{3}]$

Step 1.2

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

$\frac{1}{1 + u^{2}} \frac{d}{d x} [\frac{x}{3}]$

Step 1.3

Replace all occurrences of $u$ with $\frac{x}{3}$ .

$\frac{1}{1 + {(\frac{x}{3})}^{2}} \frac{d}{d x} [\frac{x}{3}]$

Step 2

Differentiate.

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Step 2.1

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{x}{3}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [x]$ .

$\frac{1}{1 + {(\frac{x}{3})}^{2}} (\frac{1}{3} \frac{d}{d x} [x])$

Step 2.2

Multiply $\frac{1}{3}$ by $\frac{1}{1 + {(\frac{x}{3})}^{2}}$ .

$\frac{1}{3 (1 + {(\frac{x}{3})}^{2})} \frac{d}{d x} [x]$

Step 2.3

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

$\frac{1}{3 (1 + {(\frac{x}{3})}^{2})} \cdot 1$

Step 2.4

Multiply $\frac{1}{3 (1 + {(\frac{x}{3})}^{2})}$ by $1$ .

$\frac{1}{3 (1 + {(\frac{x}{3})}^{2})}$

Step 3

Simplify.

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Step 3.1

Apply the product rule to $\frac{x}{3}$ .

$\frac{1}{3 (1 + \frac{x^{2}}{3^{2}})}$

Step 3.2

Apply the distributive property.

$\frac{1}{3 \cdot 1 + 3 \frac{x^{2}}{3^{2}}}$

Step 3.3

Combine terms.

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Step 3.3.1

Multiply $3$ by $1$ .

$\frac{1}{3 + 3 \frac{x^{2}}{3^{2}}}$

Step 3.3.2

Raise $3$ to the power of $2$ .

$\frac{1}{3 + 3 \frac{x^{2}}{9}}$

Step 3.3.3

Combine $3$ and $\frac{x^{2}}{9}$ .

$\frac{1}{3 + \frac{3 x^{2}}{9}}$

Step 3.3.4

Cancel the common factor of $3$ and $9$ .

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Step 3.3.4.1

Factor $3$ out of $3 x^{2}$ .

$\frac{1}{3 + \frac{3 (x^{2})}{9}}$

Step 3.3.4.2

Cancel the common factors.

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Step 3.3.4.2.1

Factor $3$ out of $9$ .

$\frac{1}{3 + \frac{3 x^{2}}{3 \cdot 3}}$

Step 3.3.4.2.2

Cancel the common factor.

$\frac{1}{3 + \frac{3 x^{2}}{3 \cdot 3}}$

Step 3.3.4.2.3

Rewrite the expression.

$\frac{1}{3 + \frac{x^{2}}{3}}$

Step 3.4

Reorder terms.

$\frac{1}{\frac{x^{2}}{3} + 3}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

To write $3$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{\frac{x^{2}}{3} + 3 \cdot \frac{3}{3}}$

Step 3.5.2

Combine $3$ and $\frac{3}{3}$ .

$\frac{1}{\frac{x^{2}}{3} + \frac{3 \cdot 3}{3}}$

Step 3.5.3

Combine the numerators over the common denominator.

$\frac{1}{\frac{x^{2} + 3 \cdot 3}{3}}$

Step 3.5.4

Multiply $3$ by $3$ .

$\frac{1}{\frac{x^{2} + 9}{3}}$

Step 3.6

Multiply the numerator by the reciprocal of the denominator.

$1 \frac{3}{x^{2} + 9}$

Step 3.7

Multiply $\frac{3}{x^{2} + 9}$ by $1$ .

$\frac{3}{x^{2} + 9}$