Calculus Examples

$y = \frac{1}{4} \cdot \arctan (\frac{4 \sin (x)}{3 + 5 \cos (x)})$

Step 1

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{1}{4} \cdot \arctan (\frac{4 \sin (x)}{3 + 5 \cos (x)})$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [\arctan (\frac{4 \sin (x)}{3 + 5 \cos (x)})]$ .

$\frac{1}{4} \frac{d}{d x} [\arctan (\frac{4 \sin (x)}{3 + 5 \cos (x)})]$

Step 2

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 $g (x) = \frac{4 \sin (x)}{3 + 5 \cos (x)}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $\frac{4 \sin (x)}{3 + 5 \cos (x)}$ .

$\frac{1}{4} (\frac{d}{d u} [\arctan (u)] \frac{d}{d x} [\frac{4 \sin (x)}{3 + 5 \cos (x)}])$

Step 2.2

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

$\frac{1}{4} (\frac{1}{1 + u^{2}} \frac{d}{d x} [\frac{4 \sin (x)}{3 + 5 \cos (x)}])$

Step 2.3

Replace all occurrences of $u$ with $\frac{4 \sin (x)}{3 + 5 \cos (x)}$ .

$\frac{1}{4} (\frac{1}{1 + {(\frac{4 \sin (x)}{3 + 5 \cos (x)})}^{2}} \frac{d}{d x} [\frac{4 \sin (x)}{3 + 5 \cos (x)}])$

Step 3

Differentiate using the Constant Multiple Rule.

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Step 3.1

Multiply $\frac{1}{1 + {(\frac{4 \sin (x)}{3 + 5 \cos (x)})}^{2}}$ by $\frac{1}{4}$ .

$\frac{1}{(1 + {(\frac{4 \sin (x)}{3 + 5 \cos (x)})}^{2}) \cdot 4} \frac{d}{d x} [\frac{4 \sin (x)}{3 + 5 \cos (x)}]$

Step 3.2

Move $4$ to the left of $1 + {(\frac{4 \sin (x)}{3 + 5 \cos (x)})}^{2}$ .

$\frac{1}{4 \cdot (1 + {(\frac{4 \sin (x)}{3 + 5 \cos (x)})}^{2})} \frac{d}{d x} [\frac{4 \sin (x)}{3 + 5 \cos (x)}]$

Step 3.3

Since $4$ is constant with respect to $x$ , the derivative of $\frac{4 \sin (x)}{3 + 5 \cos (x)}$ with respect to $x$ is $4 \frac{d}{d x} [\frac{\sin (x)}{3 + 5 \cos (x)}]$ .

$\frac{1}{4 (1 + {(\frac{4 \sin (x)}{3 + 5 \cos (x)})}^{2})} (4 \frac{d}{d x} [\frac{\sin (x)}{3 + 5 \cos (x)}])$

Step 3.4

Simplify terms.

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Step 3.4.1

Combine $4$ and $\frac{1}{4 (1 + {(\frac{4 \sin (x)}{3 + 5 \cos (x)})}^{2})}$ .

$\frac{4}{4 (1 + {(\frac{4 \sin (x)}{3 + 5 \cos (x)})}^{2})} \frac{d}{d x} [\frac{\sin (x)}{3 + 5 \cos (x)}]$

Step 3.4.2

Cancel the common factor of $4$ .

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Step 3.4.2.1

Cancel the common factor.

$\frac{4}{4 (1 + {(\frac{4 \sin (x)}{3 + 5 \cos (x)})}^{2})} \frac{d}{d x} [\frac{\sin (x)}{3 + 5 \cos (x)}]$

Step 3.4.2.2

Rewrite the expression.

$\frac{1}{1 + {(\frac{4 \sin (x)}{3 + 5 \cos (x)})}^{2}} \frac{d}{d x} [\frac{\sin (x)}{3 + 5 \cos (x)}]$

Step 4

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = \sin (x)$ and $g (x) = 3 + 5 \cos (x)$ .

$\frac{1}{1 + {(\frac{4 \sin (x)}{3 + 5 \cos (x)})}^{2}} \cdot \frac{(3 + 5 \cos (x)) \frac{d}{d x} [\sin (x)] - \sin (x) \frac{d}{d x} [3 + 5 \cos (x)]}{{(3 + 5 \cos (x))}^{2}}$

Step 5

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$\frac{1}{1 + {(\frac{4 \sin (x)}{3 + 5 \cos (x)})}^{2}} \cdot \frac{(3 + 5 \cos (x)) \cos (x) - \sin (x) \frac{d}{d x} [3 + 5 \cos (x)]}{{(3 + 5 \cos (x))}^{2}}$

Step 6

Differentiate.

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Step 6.1

By the Sum Rule, the derivative of $3 + 5 \cos (x)$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [5 \cos (x)]$ .

$\frac{1}{1 + {(\frac{4 \sin (x)}{3 + 5 \cos (x)})}^{2}} \cdot \frac{(3 + 5 \cos (x)) \cos (x) - \sin (x) (\frac{d}{d x} [3] + \frac{d}{d x} [5 \cos (x)])}{{(3 + 5 \cos (x))}^{2}}$

Step 6.2

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$\frac{1}{1 + {(\frac{4 \sin (x)}{3 + 5 \cos (x)})}^{2}} \cdot \frac{(3 + 5 \cos (x)) \cos (x) - \sin (x) (0 + \frac{d}{d x} [5 \cos (x)])}{{(3 + 5 \cos (x))}^{2}}$

Step 6.3

Add $0$ and $\frac{d}{d x} [5 \cos (x)]$ .

$\frac{1}{1 + {(\frac{4 \sin (x)}{3 + 5 \cos (x)})}^{2}} \cdot \frac{(3 + 5 \cos (x)) \cos (x) - \sin (x) \frac{d}{d x} [5 \cos (x)]}{{(3 + 5 \cos (x))}^{2}}$

Step 6.4

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

$\frac{1}{1 + {(\frac{4 \sin (x)}{3 + 5 \cos (x)})}^{2}} \cdot \frac{(3 + 5 \cos (x)) \cos (x) - \sin (x) (5 \frac{d}{d x} [\cos (x)])}{{(3 + 5 \cos (x))}^{2}}$

Step 6.5

Multiply $5$ by $- 1$ .

$\frac{1}{1 + {(\frac{4 \sin (x)}{3 + 5 \cos (x)})}^{2}} \cdot \frac{(3 + 5 \cos (x)) \cos (x) - 5 \sin (x) \frac{d}{d x} [\cos (x)]}{{(3 + 5 \cos (x))}^{2}}$

Step 7

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$\frac{1}{1 + {(\frac{4 \sin (x)}{3 + 5 \cos (x)})}^{2}} \cdot \frac{(3 + 5 \cos (x)) \cos (x) - 5 \sin (x) (- \sin (x))}{{(3 + 5 \cos (x))}^{2}}$

Step 8

Multiply $- 1$ by $- 5$ .

$\frac{1}{1 + {(\frac{4 \sin (x)}{3 + 5 \cos (x)})}^{2}} \cdot \frac{(3 + 5 \cos (x)) \cos (x) + 5 \sin (x) \sin (x)}{{(3 + 5 \cos (x))}^{2}}$

Step 9

Raise $\sin (x)$ to the power of $1$ .

$\frac{1}{1 + {(\frac{4 \sin (x)}{3 + 5 \cos (x)})}^{2}} \cdot \frac{(3 + 5 \cos (x)) \cos (x) + 5 (\sin^{1} (x) \sin (x))}{{(3 + 5 \cos (x))}^{2}}$

Step 10

Raise $\sin (x)$ to the power of $1$ .

$\frac{1}{1 + {(\frac{4 \sin (x)}{3 + 5 \cos (x)})}^{2}} \cdot \frac{(3 + 5 \cos (x)) \cos (x) + 5 (\sin^{1} (x) \sin^{1} (x))}{{(3 + 5 \cos (x))}^{2}}$

Step 11

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{1 + {(\frac{4 \sin (x)}{3 + 5 \cos (x)})}^{2}} \cdot \frac{(3 + 5 \cos (x)) \cos (x) + 5 {\sin (x)}^{1 + 1}}{{(3 + 5 \cos (x))}^{2}}$

Step 12

Add $1$ and $1$ .

$\frac{1}{1 + {(\frac{4 \sin (x)}{3 + 5 \cos (x)})}^{2}} \cdot \frac{(3 + 5 \cos (x)) \cos (x) + 5 \sin^{2} (x)}{{(3 + 5 \cos (x))}^{2}}$

Step 13

Multiply $\frac{1}{1 + {(\frac{4 \sin (x)}{3 + 5 \cos (x)})}^{2}}$ by $\frac{(3 + 5 \cos (x)) \cos (x) + 5 \sin^{2} (x)}{{(3 + 5 \cos (x))}^{2}}$ .

$\frac{(3 + 5 \cos (x)) \cos (x) + 5 \sin^{2} (x)}{(1 + {(\frac{4 \sin (x)}{3 + 5 \cos (x)})}^{2}) {(3 + 5 \cos (x))}^{2}}$

Step 14

Simplify.

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Step 14.1

Apply the product rule to $\frac{4 \sin (x)}{3 + 5 \cos (x)}$ .

$\frac{(3 + 5 \cos (x)) \cos (x) + 5 \sin^{2} (x)}{(1 + \frac{{(4 \sin (x))}^{2}}{{(3 + 5 \cos (x))}^{2}}) {(3 + 5 \cos (x))}^{2}}$

Step 14.2

Apply the product rule to $4 \sin (x)$ .

$\frac{(3 + 5 \cos (x)) \cos (x) + 5 \sin^{2} (x)}{(1 + \frac{4^{2} \sin^{2} (x)}{{(3 + 5 \cos (x))}^{2}}) {(3 + 5 \cos (x))}^{2}}$

Step 14.3

Apply the distributive property.

$\frac{3 \cos (x) + 5 \cos (x) \cos (x) + 5 \sin^{2} (x)}{(1 + \frac{4^{2} \sin^{2} (x)}{{(3 + 5 \cos (x))}^{2}}) {(3 + 5 \cos (x))}^{2}}$

Step 14.4

Simplify the numerator.

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Step 14.4.1

Multiply $5 \cos (x) \cos (x)$ .

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Step 14.4.1.1

Raise $\cos (x)$ to the power of $1$ .

$\frac{3 \cos (x) + 5 (\cos^{1} (x) \cos (x)) + 5 \sin^{2} (x)}{(1 + \frac{4^{2} \sin^{2} (x)}{{(3 + 5 \cos (x))}^{2}}) {(3 + 5 \cos (x))}^{2}}$

Step 14.4.1.2

Raise $\cos (x)$ to the power of $1$ .

$\frac{3 \cos (x) + 5 (\cos^{1} (x) \cos^{1} (x)) + 5 \sin^{2} (x)}{(1 + \frac{4^{2} \sin^{2} (x)}{{(3 + 5 \cos (x))}^{2}}) {(3 + 5 \cos (x))}^{2}}$

Step 14.4.1.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 \cos (x) + 5 {\cos (x)}^{1 + 1} + 5 \sin^{2} (x)}{(1 + \frac{4^{2} \sin^{2} (x)}{{(3 + 5 \cos (x))}^{2}}) {(3 + 5 \cos (x))}^{2}}$

Step 14.4.1.4

Add $1$ and $1$ .

$\frac{3 \cos (x) + 5 \cos^{2} (x) + 5 \sin^{2} (x)}{(1 + \frac{4^{2} \sin^{2} (x)}{{(3 + 5 \cos (x))}^{2}}) {(3 + 5 \cos (x))}^{2}}$

Step 14.4.2

Factor $5$ out of $5 \cos^{2} (x)$ .

$\frac{3 \cos (x) + 5 (\cos^{2} (x)) + 5 \sin^{2} (x)}{(1 + \frac{4^{2} \sin^{2} (x)}{{(3 + 5 \cos (x))}^{2}}) {(3 + 5 \cos (x))}^{2}}$

Step 14.4.3

Factor $5$ out of $5 \sin^{2} (x)$ .

$\frac{3 \cos (x) + 5 (\cos^{2} (x)) + 5 (\sin^{2} (x))}{(1 + \frac{4^{2} \sin^{2} (x)}{{(3 + 5 \cos (x))}^{2}}) {(3 + 5 \cos (x))}^{2}}$

Step 14.4.4

Factor $5$ out of $5 (\cos^{2} (x)) + 5 (\sin^{2} (x))$ .

$\frac{3 \cos (x) + 5 (\cos^{2} (x) + \sin^{2} (x))}{(1 + \frac{4^{2} \sin^{2} (x)}{{(3 + 5 \cos (x))}^{2}}) {(3 + 5 \cos (x))}^{2}}$

Step 14.4.5

Rearrange terms.

$\frac{3 \cos (x) + 5 (\sin^{2} (x) + \cos^{2} (x))}{(1 + \frac{4^{2} \sin^{2} (x)}{{(3 + 5 \cos (x))}^{2}}) {(3 + 5 \cos (x))}^{2}}$

Step 14.4.6

Apply pythagorean identity.

$\frac{3 \cos (x) + 5 \cdot 1}{(1 + \frac{4^{2} \sin^{2} (x)}{{(3 + 5 \cos (x))}^{2}}) {(3 + 5 \cos (x))}^{2}}$

Step 14.4.7

Multiply $5$ by $1$ .

$\frac{3 \cos (x) + 5}{(1 + \frac{4^{2} \sin^{2} (x)}{{(3 + 5 \cos (x))}^{2}}) {(3 + 5 \cos (x))}^{2}}$

Step 14.5

Raise $4$ to the power of $2$ .

$\frac{3 \cos (x) + 5}{(1 + \frac{16 \sin^{2} (x)}{{(3 + 5 \cos (x))}^{2}}) {(3 + 5 \cos (x))}^{2}}$

Step 14.6

Reorder terms.

$\frac{5 + 3 \cos (x)}{{(3 + 5 \cos (x))}^{2} (1 + \frac{16 \sin^{2} (x)}{{(3 + 5 \cos (x))}^{2}})}$