Calculus Examples

$y = \arctan (\frac{x}{6})$

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 $g (x) = \frac{x}{6}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{x}{6}$ .

$\frac{d}{d u} [\arctan (u)] \frac{d}{d x} [\frac{x}{6}]$

Step 1.2

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

$\frac{1}{1 + u^{2}} \frac{d}{d x} [\frac{x}{6}]$

Step 1.3

Replace all occurrences of $u$ with $\frac{x}{6}$ .

$\frac{1}{1 + {(\frac{x}{6})}^{2}} \frac{d}{d x} [\frac{x}{6}]$

Step 2

Differentiate.

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Step 2.1

Since $\frac{1}{6}$ is constant with respect to $x$ , the derivative of $\frac{x}{6}$ with respect to $x$ is $\frac{1}{6} \frac{d}{d x} [x]$ .

$\frac{1}{1 + {(\frac{x}{6})}^{2}} (\frac{1}{6} \frac{d}{d x} [x])$

Step 2.2

Multiply $\frac{1}{6}$ by $\frac{1}{1 + {(\frac{x}{6})}^{2}}$ .

$\frac{1}{6 (1 + {(\frac{x}{6})}^{2})} \frac{d}{d x} [x]$

Step 2.3

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

$\frac{1}{6 (1 + {(\frac{x}{6})}^{2})} \cdot 1$

Step 2.4

Multiply $\frac{1}{6 (1 + {(\frac{x}{6})}^{2})}$ by $1$ .

$\frac{1}{6 (1 + {(\frac{x}{6})}^{2})}$

Step 3

Simplify.

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Step 3.1

Apply the product rule to $\frac{x}{6}$ .

$\frac{1}{6 (1 + \frac{x^{2}}{6^{2}})}$

Step 3.2

Apply the distributive property.

$\frac{1}{6 \cdot 1 + 6 \frac{x^{2}}{6^{2}}}$

Step 3.3

Combine terms.

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Step 3.3.1

Multiply $6$ by $1$ .

$\frac{1}{6 + 6 \frac{x^{2}}{6^{2}}}$

Step 3.3.2

Raise $6$ to the power of $2$ .

$\frac{1}{6 + 6 \frac{x^{2}}{36}}$

Step 3.3.3

Combine $6$ and $\frac{x^{2}}{36}$ .

$\frac{1}{6 + \frac{6 x^{2}}{36}}$

Step 3.3.4

Cancel the common factor of $6$ and $36$ .

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Step 3.3.4.1

Factor $6$ out of $6 x^{2}$ .

$\frac{1}{6 + \frac{6 (x^{2})}{36}}$

Step 3.3.4.2

Cancel the common factors.

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Step 3.3.4.2.1

Factor $6$ out of $36$ .

$\frac{1}{6 + \frac{6 x^{2}}{6 \cdot 6}}$

Step 3.3.4.2.2

Cancel the common factor.

$\frac{1}{6 + \frac{6 x^{2}}{6 \cdot 6}}$

Step 3.3.4.2.3

Rewrite the expression.

$\frac{1}{6 + \frac{x^{2}}{6}}$

Step 3.4

Reorder terms.

$\frac{1}{\frac{x^{2}}{6} + 6}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

To write $6$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{1}{\frac{x^{2}}{6} + 6 \cdot \frac{6}{6}}$

Step 3.5.2

Combine $6$ and $\frac{6}{6}$ .

$\frac{1}{\frac{x^{2}}{6} + \frac{6 \cdot 6}{6}}$

Step 3.5.3

Combine the numerators over the common denominator.

$\frac{1}{\frac{x^{2} + 6 \cdot 6}{6}}$

Step 3.5.4

Multiply $6$ by $6$ .

$\frac{1}{\frac{x^{2} + 36}{6}}$

Step 3.6

Multiply the numerator by the reciprocal of the denominator.

$1 \frac{6}{x^{2} + 36}$

Step 3.7

Multiply $\frac{6}{x^{2} + 36}$ by $1$ .

$\frac{6}{x^{2} + 36}$