Calculus Examples

${(\frac{\cos (2 x)}{x})}^{3}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = \frac{\cos (2 x)}{x}$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\frac{\cos (2 x)}{x}$ .

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

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 3$ .

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

Step 1.3

Replace all occurrences of $u_{1}$ with $\frac{\cos (2 x)}{x}$ .

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

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 $f (x) = \cos (2 x)$ and $g (x) = x$ .

$3 {(\frac{\cos (2 x)}{x})}^{2} \frac{x \frac{d}{d x} [\cos (2 x)] - \cos (2 x) \frac{d}{d x} [x]}{x^{2}}$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 2 x$ .

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Step 3.1

To apply the Chain Rule, set $u_{2}$ as $2 x$ .

$3 {(\frac{\cos (2 x)}{x})}^{2} \frac{x (\frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d x} [2 x]) - \cos (2 x) \frac{d}{d x} [x]}{x^{2}}$

Step 3.2

The derivative of $\cos (u_{2})$ with respect to $u_{2}$ is $- \sin (u_{2})$ .

$3 {(\frac{\cos (2 x)}{x})}^{2} \frac{x (- \sin (u_{2}) \frac{d}{d x} [2 x]) - \cos (2 x) \frac{d}{d x} [x]}{x^{2}}$

Step 3.3

Replace all occurrences of $u_{2}$ with $2 x$ .

$3 {(\frac{\cos (2 x)}{x})}^{2} \frac{x (- \sin (2 x) \frac{d}{d x} [2 x]) - \cos (2 x) \frac{d}{d x} [x]}{x^{2}}$

Step 4

Differentiate.

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Step 4.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$3 {(\frac{\cos (2 x)}{x})}^{2} \frac{x (- \sin (2 x) (2 \frac{d}{d x} [x])) - \cos (2 x) \frac{d}{d x} [x]}{x^{2}}$

Step 4.2

Multiply $2$ by $- 1$ .

$3 {(\frac{\cos (2 x)}{x})}^{2} \frac{x (- 2 \sin (2 x) \frac{d}{d x} [x]) - \cos (2 x) \frac{d}{d x} [x]}{x^{2}}$

Step 4.3

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

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

Step 4.4

Multiply $- 2$ by $1$ .

$3 {(\frac{\cos (2 x)}{x})}^{2} \frac{x (- 2 \sin (2 x)) - \cos (2 x) \frac{d}{d x} [x]}{x^{2}}$

Step 4.5

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

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

Step 4.6

Combine fractions.

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Step 4.6.1

Multiply $- 1$ by $1$ .

$3 {(\frac{\cos (2 x)}{x})}^{2} \frac{x (- 2 \sin (2 x)) - \cos (2 x)}{x^{2}}$

Step 4.6.2

Combine $\frac{x (- 2 \sin (2 x)) - \cos (2 x)}{x^{2}}$ and $3$ .

$\frac{(x (- 2 \sin (2 x)) - \cos (2 x)) \cdot 3}{x^{2}} {(\frac{\cos (2 x)}{x})}^{2}$

Step 4.6.3

Move $3$ to the left of $x (- 2 \sin (2 x)) - \cos (2 x)$ .

$\frac{3 \cdot (x (- 2 \sin (2 x)) - \cos (2 x))}{x^{2}} {(\frac{\cos (2 x)}{x})}^{2}$

Step 5

Simplify.

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Step 5.1

Apply the product rule to $\frac{\cos (2 x)}{x}$ .

$\frac{3 (x (- 2 \sin (2 x)) - \cos (2 x))}{x^{2}} \cdot \frac{\cos^{2} (2 x)}{x^{2}}$

Step 5.2

Apply the distributive property.

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

Step 5.3

Combine terms.

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Step 5.3.1

Multiply $- 2$ by $3$ .

$\frac{- 6 (x (\sin (2 x))) + 3 (- \cos (2 x))}{x^{2}} \cdot \frac{\cos^{2} (2 x)}{x^{2}}$

Step 5.3.2

Multiply $- 1$ by $3$ .

$\frac{- 6 x \sin (2 x) - 3 \cos (2 x)}{x^{2}} \cdot \frac{\cos^{2} (2 x)}{x^{2}}$

Step 5.3.3

Multiply $\frac{- 6 x \sin (2 x) - 3 \cos (2 x)}{x^{2}}$ by $\frac{\cos^{2} (2 x)}{x^{2}}$ .

$\frac{(- 6 x \sin (2 x) - 3 \cos (2 x)) \cos^{2} (2 x)}{x^{2} x^{2}}$

Step 5.3.4

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 5.3.4.1

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

$\frac{(- 6 x \sin (2 x) - 3 \cos (2 x)) \cos^{2} (2 x)}{x^{2 + 2}}$

Step 5.3.4.2

Add $2$ and $2$ .

$\frac{(- 6 x \sin (2 x) - 3 \cos (2 x)) \cos^{2} (2 x)}{x^{4}}$

Step 5.4

Reorder terms.

$\frac{\cos^{2} (2 x) (- 6 x \sin (2 x) - 3 \cos (2 x))}{x^{4}}$

Step 5.5

Factor $3$ out of $- 6 x \sin (2 x) - 3 \cos (2 x)$ .

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Step 5.5.1

Factor $3$ out of $- 6 x \sin (2 x)$ .

$\frac{\cos^{2} (2 x) (3 (- 2 x \sin (2 x)) - 3 \cos (2 x))}{x^{4}}$

Step 5.5.2

Factor $3$ out of $- 3 \cos (2 x)$ .

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

Step 5.5.3

Factor $3$ out of $3 (- 2 x \sin (2 x)) + 3 (- \cos (2 x))$ .

$\frac{\cos^{2} (2 x) (3 (- 2 x \sin (2 x) - \cos (2 x)))}{x^{4}}$

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

Step 5.6

Move $3$ to the left of $\cos^{2} (2 x)$ .

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

Step 5.7

Factor $- 1$ out of $- 2 x \sin (2 x)$ .

$\frac{3 \cos^{2} (2 x) (- (2 x \sin (2 x)) - \cos (2 x))}{x^{4}}$

Step 5.8

Factor $- 1$ out of $- \cos (2 x)$ .

$\frac{3 \cos^{2} (2 x) (- (2 x \sin (2 x)) - (\cos (2 x)))}{x^{4}}$

Step 5.9

Factor $- 1$ out of $- (2 x \sin (2 x)) - (\cos (2 x))$ .

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

Step 5.10

Rewrite $- (2 x \sin (2 x) + \cos (2 x))$ as $- 1 (2 x \sin (2 x) + \cos (2 x))$ .

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

Step 5.11

Move the negative in front of the fraction.

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

Step 5.12

Reorder factors in $- \frac{(3 \cos^{2} (2 x)) (2 x \sin (2 x) + \cos (2 x))}{x^{4}}$ .

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