Calculus Examples

$\arctan (\frac{6 x}{5})$

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 $g (x) = \frac{6 x}{5}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{6 x}{5}$ .

$\frac{d}{d u} [\arctan (u)] \frac{d}{d x} [\frac{6 x}{5}]$

Step 1.2

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

$\frac{1}{1 + u^{2}} \frac{d}{d x} [\frac{6 x}{5}]$

Step 1.3

Replace all occurrences of $u$ with $\frac{6 x}{5}$ .

$\frac{1}{1 + {(\frac{6 x}{5})}^{2}} \frac{d}{d x} [\frac{6 x}{5}]$

Step 2

Differentiate.

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Step 2.1

Since $\frac{6}{5}$ is constant with respect to $x$ , the derivative of $\frac{6 x}{5}$ with respect to $x$ is $\frac{6}{5} \frac{d}{d x} [x]$ .

$\frac{1}{1 + {(\frac{6 x}{5})}^{2}} (\frac{6}{5} \frac{d}{d x} [x])$

Step 2.2

Multiply $\frac{6}{5}$ by $\frac{1}{1 + {(\frac{6 x}{5})}^{2}}$ .

$\frac{6}{5 (1 + {(\frac{6 x}{5})}^{2})} \frac{d}{d x} [x]$

Step 2.3

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

$\frac{6}{5 (1 + {(\frac{6 x}{5})}^{2})} \cdot 1$

Step 2.4

Multiply $\frac{6}{5 (1 + {(\frac{6 x}{5})}^{2})}$ by $1$ .

$\frac{6}{5 (1 + {(\frac{6 x}{5})}^{2})}$

Step 3

Simplify.

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Step 3.1

Apply the product rule to $\frac{6 x}{5}$ .

$\frac{6}{5 (1 + \frac{{(6 x)}^{2}}{5^{2}})}$

Step 3.2

Apply the product rule to $6 x$ .

$\frac{6}{5 (1 + \frac{6^{2} x^{2}}{5^{2}})}$

Step 3.3

Apply the distributive property.

$\frac{6}{5 \cdot 1 + 5 \frac{6^{2} x^{2}}{5^{2}}}$

Step 3.4

Combine terms.

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Step 3.4.1

Multiply $5$ by $1$ .

$\frac{6}{5 + 5 \frac{6^{2} x^{2}}{5^{2}}}$

Step 3.4.2

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

$\frac{6}{5 + 5 \frac{36 x^{2}}{5^{2}}}$

Step 3.4.3

Raise $5$ to the power of $2$ .

$\frac{6}{5 + 5 \frac{36 x^{2}}{25}}$

Step 3.4.4

Combine $5$ and $\frac{36 x^{2}}{25}$ .

$\frac{6}{5 + \frac{5 (36 x^{2})}{25}}$

Step 3.4.5

Multiply $36$ by $5$ .

$\frac{6}{5 + \frac{180 x^{2}}{25}}$

Step 3.4.6

Cancel the common factor of $180$ and $25$ .

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Step 3.4.6.1

Factor $5$ out of $180 x^{2}$ .

$\frac{6}{5 + \frac{5 (36 x^{2})}{25}}$

Step 3.4.6.2

Cancel the common factors.

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Step 3.4.6.2.1

Factor $5$ out of $25$ .

$\frac{6}{5 + \frac{5 (36 x^{2})}{5 (5)}}$

Step 3.4.6.2.2

Cancel the common factor.

$\frac{6}{5 + \frac{5 (36 x^{2})}{5 \cdot 5}}$

Step 3.4.6.2.3

Rewrite the expression.

$\frac{6}{5 + \frac{36 x^{2}}{5}}$

Step 3.5

Reorder terms.

$\frac{6}{\frac{36 x^{2}}{5} + 5}$

Step 3.6

Simplify the denominator.

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Step 3.6.1

To write $5$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{6}{\frac{36 x^{2}}{5} + 5 \cdot \frac{5}{5}}$

Step 3.6.2

Combine $5$ and $\frac{5}{5}$ .

$\frac{6}{\frac{36 x^{2}}{5} + \frac{5 \cdot 5}{5}}$

Step 3.6.3

Combine the numerators over the common denominator.

$\frac{6}{\frac{36 x^{2} + 5 \cdot 5}{5}}$

Step 3.6.4

Multiply $5$ by $5$ .

$\frac{6}{\frac{36 x^{2} + 25}{5}}$

Step 3.7

Multiply the numerator by the reciprocal of the denominator.

$6 \frac{5}{36 x^{2} + 25}$

Step 3.8

Multiply $6 \frac{5}{36 x^{2} + 25}$ .

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Step 3.8.1

Combine $6$ and $\frac{5}{36 x^{2} + 25}$ .

$\frac{6 \cdot 5}{36 x^{2} + 25}$

Step 3.8.2

Multiply $6$ by $5$ .

$\frac{30}{36 x^{2} + 25}$