Calculus Examples

$\frac{8 x}{\sin (x) + \cos (x)}$

Step 1

Since $8$ is constant with respect to $x$ , the derivative of $\frac{8 x}{\sin (x) + \cos (x)}$ with respect to $x$ is $8 \frac{d}{d x} [\frac{x}{\sin (x) + \cos (x)}]$ .

$8 \frac{d}{d x} [\frac{x}{\sin (x) + \cos (x)}]$

Step 2

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 $f (x) = x$ and $g (x) = \sin (x) + \cos (x)$ .

$8 \frac{(\sin (x) + \cos (x)) \frac{d}{d x} [x] - x \frac{d}{d x} [\sin (x) + \cos (x)]}{{(\sin (x) + \cos (x))}^{2}}$

Step 3

Differentiate.

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Step 3.1

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

$8 \frac{(\sin (x) + \cos (x)) \cdot 1 - x \frac{d}{d x} [\sin (x) + \cos (x)]}{{(\sin (x) + \cos (x))}^{2}}$

Step 3.2

Multiply $\sin (x) + \cos (x)$ by $1$ .

$8 \frac{\sin (x) + \cos (x) - x \frac{d}{d x} [\sin (x) + \cos (x)]}{{(\sin (x) + \cos (x))}^{2}}$

Step 3.3

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

$8 \frac{\sin (x) + \cos (x) - x (\frac{d}{d x} [\sin (x)] + \frac{d}{d x} [\cos (x)])}{{(\sin (x) + \cos (x))}^{2}}$

Step 4

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

$8 \frac{\sin (x) + \cos (x) - x (\cos (x) + \frac{d}{d x} [\cos (x)])}{{(\sin (x) + \cos (x))}^{2}}$

Step 5

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

$8 \frac{\sin (x) + \cos (x) - x (\cos (x) - \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$

Step 6

Combine $8$ and $\frac{\sin (x) + \cos (x) - x (\cos (x) - \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$ .

$\frac{8 (\sin (x) + \cos (x) - x (\cos (x) - \sin (x)))}{{(\sin (x) + \cos (x))}^{2}}$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$\frac{8 (\sin (x) + \cos (x) - x \cos (x) - x (- \sin (x)))}{{(\sin (x) + \cos (x))}^{2}}$

Step 7.2

Apply the distributive property.

$\frac{8 \sin (x) + 8 \cos (x) + 8 (- x \cos (x)) + 8 (- x (- \sin (x)))}{{(\sin (x) + \cos (x))}^{2}}$

Step 7.3

Simplify each term.

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Step 7.3.1

Multiply $- 1$ by $8$ .

$\frac{8 \sin (x) + 8 \cos (x) - 8 (x \cos (x)) + 8 (- x (- \sin (x)))}{{(\sin (x) + \cos (x))}^{2}}$

Step 7.3.2

Multiply $- x (- \sin (x))$ .

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Step 7.3.2.1

Multiply $- 1$ by $- 1$ .

$\frac{8 \sin (x) + 8 \cos (x) - 8 x \cos (x) + 8 (1 x \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$

Step 7.3.2.2

Multiply $x$ by $1$ .

$\frac{8 \sin (x) + 8 \cos (x) - 8 x \cos (x) + 8 (x \sin (x))}{{(\sin (x) + \cos (x))}^{2}}$

$\frac{8 \sin (x) + 8 \cos (x) - 8 x \cos (x) + 8 x \sin (x)}{{(\sin (x) + \cos (x))}^{2}}$

Step 7.4

Reorder terms.

$\frac{- 8 x \cos (x) + 8 x \sin (x) + 8 \sin (x) + 8 \cos (x)}{{(\sin (x) + \cos (x))}^{2}}$

Step 7.5

Factor $8$ out of $- 8 x \cos (x) + 8 x \sin (x) + 8 \sin (x) + 8 \cos (x)$ .

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Step 7.5.1

Factor $8$ out of $- 8 x \cos (x)$ .

$\frac{8 (- x \cos (x)) + 8 x \sin (x) + 8 \sin (x) + 8 \cos (x)}{{(\sin (x) + \cos (x))}^{2}}$

Step 7.5.2

Factor $8$ out of $8 x \sin (x)$ .

$\frac{8 (- x \cos (x)) + 8 (x \sin (x)) + 8 \sin (x) + 8 \cos (x)}{{(\sin (x) + \cos (x))}^{2}}$

Step 7.5.3

Factor $8$ out of $8 \sin (x)$ .

$\frac{8 (- x \cos (x)) + 8 (x \sin (x)) + 8 (\sin (x)) + 8 \cos (x)}{{(\sin (x) + \cos (x))}^{2}}$

Step 7.5.4

Factor $8$ out of $8 \cos (x)$ .

$\frac{8 (- x \cos (x)) + 8 (x \sin (x)) + 8 (\sin (x)) + 8 (\cos (x))}{{(\sin (x) + \cos (x))}^{2}}$

Step 7.5.5

Factor $8$ out of $8 (- x \cos (x)) + 8 (x \sin (x))$ .

$\frac{8 (- x \cos (x) + x \sin (x)) + 8 (\sin (x)) + 8 (\cos (x))}{{(\sin (x) + \cos (x))}^{2}}$

Step 7.5.6

Factor $8$ out of $8 (- x \cos (x) + x \sin (x)) + 8 (\sin (x))$ .

$\frac{8 (- x \cos (x) + x \sin (x) + \sin (x)) + 8 (\cos (x))}{{(\sin (x) + \cos (x))}^{2}}$

Step 7.5.7

Factor $8$ out of $8 (- x \cos (x) + x \sin (x) + \sin (x)) + 8 (\cos (x))$ .

$\frac{8 (- x \cos (x) + x \sin (x) + \sin (x) + \cos (x))}{{(\sin (x) + \cos (x))}^{2}}$

Step 7.6

Factor $- 1$ out of $- x \cos (x)$ .

$\frac{8 (- (x \cos (x)) + x \sin (x) + \sin (x) + \cos (x))}{{(\sin (x) + \cos (x))}^{2}}$

Step 7.7

Factor $- 1$ out of $x \sin (x)$ .

$\frac{8 (- (x \cos (x)) - (- x \sin (x)) + \sin (x) + \cos (x))}{{(\sin (x) + \cos (x))}^{2}}$

Step 7.8

Factor $- 1$ out of $- (x \cos (x)) - (- x \sin (x))$ .

$\frac{8 (- (x \cos (x) - x \sin (x)) + \sin (x) + \cos (x))}{{(\sin (x) + \cos (x))}^{2}}$

Step 7.9

Factor $- 1$ out of $\sin (x)$ .

$\frac{8 (- (x \cos (x) - x \sin (x)) - 1 (- \sin (x)) + \cos (x))}{{(\sin (x) + \cos (x))}^{2}}$

Step 7.10

Factor $- 1$ out of $- (x \cos (x) - x \sin (x)) - 1 (- \sin (x))$ .

$\frac{8 (- (x \cos (x) - x \sin (x) - \sin (x)) + \cos (x))}{{(\sin (x) + \cos (x))}^{2}}$

Step 7.11

Factor $- 1$ out of $\cos (x)$ .

$\frac{8 (- (x \cos (x) - x \sin (x) - \sin (x)) - 1 (- \cos (x)))}{{(\sin (x) + \cos (x))}^{2}}$

Step 7.12

Factor $- 1$ out of $- (x \cos (x) - x \sin (x) - \sin (x)) - 1 (- \cos (x))$ .

$\frac{8 (- (x \cos (x) - x \sin (x) - \sin (x) - \cos (x)))}{{(\sin (x) + \cos (x))}^{2}}$

Step 7.13

Rewrite $- (x \cos (x) - x \sin (x) - \sin (x) - \cos (x))$ as $- 1 (x \cos (x) - x \sin (x) - \sin (x) - \cos (x))$ .

$\frac{8 (- 1 (x \cos (x) - x \sin (x) - \sin (x) - \cos (x)))}{{(\sin (x) + \cos (x))}^{2}}$

Step 7.14

Move the negative in front of the fraction.

$- \frac{8 (x \cos (x) - x \sin (x) - \sin (x) - \cos (x))}{{(\sin (x) + \cos (x))}^{2}}$