Calculus Examples

$f' (x) = \frac{d}{d x} \cdot 8 \cos (2 x)$

Step 1

Find the first derivative.

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Step 1.1

Differentiate using the Constant Multiple Rule.

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Step 1.1.1

Cancel the common factor of $d$ .

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Step 1.1.1.1

Cancel the common factor.

$\frac{d}{d x} [\frac{d}{d x} \cdot (8 \cos (2 x))]$

Step 1.1.1.2

Rewrite the expression.

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

Step 1.1.2

Combine fractions.

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Step 1.1.2.1

Combine $8$ and $\frac{1}{x}$ .

$\frac{d}{d x} [\frac{8}{x} \cdot \cos (2 x)]$

Step 1.1.2.2

Combine $\frac{8}{x}$ and $\cos (2 x)$ .

$\frac{d}{d x} [\frac{8 \cos (2 x)}{x}]$

Step 1.1.3

Since $8$ is constant with respect to $x$ , the derivative of $\frac{8 \cos (2 x)}{x}$ with respect to $x$ is $8 \frac{d}{d x} [\frac{\cos (2 x)}{x}]$ .

$8 \frac{d}{d x} [\frac{\cos (2 x)}{x}]$

Step 1.2

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 $g (x) = x$ .

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

Step 1.3

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 1.3.1

To apply the Chain Rule, set $u$ as $2 x$ .

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

Step 1.3.2

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

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

Step 1.3.3

Replace all occurrences of $u$ with $2 x$ .

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

Step 1.4

Differentiate.

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Step 1.4.1

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

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

Step 1.4.2

Multiply $2$ by $- 1$ .

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

Step 1.4.3

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

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

Step 1.4.4

Multiply $- 2$ by $1$ .

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

Step 1.4.5

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

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

Step 1.4.6

Combine fractions.

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Step 1.4.6.1

Multiply $- 1$ by $1$ .

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

Step 1.4.6.2

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

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

Step 1.5

Simplify.

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Step 1.5.1

Apply the distributive property.

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

Step 1.5.2

Simplify each term.

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Step 1.5.2.1

Rewrite using the commutative property of multiplication.

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

Step 1.5.2.2

Multiply $- 2$ by $8$ .

$\frac{- 16 (x \sin (2 x)) + 8 (- \cos (2 x))}{x^{2}}$

Step 1.5.2.3

Multiply $- 1$ by $8$ .

$\frac{- 16 x \sin (2 x) - 8 \cos (2 x)}{x^{2}}$

Step 1.5.3

Factor $8$ out of $- 16 x \sin (2 x) - 8 \cos (2 x)$ .

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Step 1.5.3.1

Factor $8$ out of $- 16 x \sin (2 x)$ .

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

Step 1.5.3.2

Factor $8$ out of $- 8 \cos (2 x)$ .

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

Step 1.5.3.3

Factor $8$ out of $8 (- 2 x \sin (2 x)) + 8 (- \cos (2 x))$ .

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

Step 1.5.4

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

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

Step 1.5.5

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

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

Step 1.5.6

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

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

Step 1.5.7

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

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

Step 1.5.8

Move the negative in front of the fraction.

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

Step 2

Find the second derivative.

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Step 2.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- \frac{8 (2 x \sin (2 x) + \cos (2 x))}{x^{2}}$ with respect to $x$ is $- 8 \frac{d}{d x} [\frac{2 x \sin (2 x) + \cos (2 x)}{x^{2}}]$ .

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

Step 2.2

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

Step 2.3

Differentiate.

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Step 2.3.1

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 2.3.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.3.1.2

Multiply $2$ by $2$ .

$- 8 \frac{x^{2} \frac{d}{d x} [2 x \sin (2 x) + \cos (2 x)] - (2 x \sin (2 x) + \cos (2 x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.3.2

By the Sum Rule, the derivative of $2 x \sin (2 x) + \cos (2 x)$ with respect to $x$ is $\frac{d}{d x} [2 x \sin (2 x)] + \frac{d}{d x} [\cos (2 x)]$ .

$- 8 \frac{x^{2} (\frac{d}{d x} [2 x \sin (2 x)] + \frac{d}{d x} [\cos (2 x)]) - (2 x \sin (2 x) + \cos (2 x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.3.3

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

$- 8 \frac{x^{2} (2 \frac{d}{d x} [x \sin (2 x)] + \frac{d}{d x} [\cos (2 x)]) - (2 x \sin (2 x) + \cos (2 x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.4

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

$- 8 \frac{x^{2} (2 (x \frac{d}{d x} [\sin (2 x)] + \sin (2 x) \frac{d}{d x} [x]) + \frac{d}{d x} [\cos (2 x)]) - (2 x \sin (2 x) + \cos (2 x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.5

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

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Step 2.5.1

To apply the Chain Rule, set $u_{1}$ as $2 x$ .

$- 8 \frac{x^{2} (2 (x (\frac{d}{d u_{1}} [\sin (u_{1})] \frac{d}{d x} [2 x]) + \sin (2 x) \frac{d}{d x} [x]) + \frac{d}{d x} [\cos (2 x)]) - (2 x \sin (2 x) + \cos (2 x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.5.2

The derivative of $\sin (u_{1})$ with respect to $u_{1}$ is $\cos (u_{1})$ .

$- 8 \frac{x^{2} (2 (x (\cos (u_{1}) \frac{d}{d x} [2 x]) + \sin (2 x) \frac{d}{d x} [x]) + \frac{d}{d x} [\cos (2 x)]) - (2 x \sin (2 x) + \cos (2 x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.5.3

Replace all occurrences of $u_{1}$ with $2 x$ .

$- 8 \frac{x^{2} (2 (x (\cos (2 x) \frac{d}{d x} [2 x]) + \sin (2 x) \frac{d}{d x} [x]) + \frac{d}{d x} [\cos (2 x)]) - (2 x \sin (2 x) + \cos (2 x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.6

Differentiate.

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Step 2.6.1

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

$- 8 \frac{x^{2} (2 (x (\cos (2 x) (2 \frac{d}{d x} [x])) + \sin (2 x) \frac{d}{d x} [x]) + \frac{d}{d x} [\cos (2 x)]) - (2 x \sin (2 x) + \cos (2 x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.6.2

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

$- 8 \frac{x^{2} (2 (x (\cos (2 x) (2 \cdot 1)) + \sin (2 x) \frac{d}{d x} [x]) + \frac{d}{d x} [\cos (2 x)]) - (2 x \sin (2 x) + \cos (2 x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.6.3

Simplify the expression.

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Step 2.6.3.1

Multiply $2$ by $1$ .

$- 8 \frac{x^{2} (2 (x (\cos (2 x) \cdot 2) + \sin (2 x) \frac{d}{d x} [x]) + \frac{d}{d x} [\cos (2 x)]) - (2 x \sin (2 x) + \cos (2 x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.6.3.2

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

$- 8 \frac{x^{2} (2 (x (2 \cdot \cos (2 x)) + \sin (2 x) \frac{d}{d x} [x]) + \frac{d}{d x} [\cos (2 x)]) - (2 x \sin (2 x) + \cos (2 x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.6.4

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

$- 8 \frac{x^{2} (2 (x (2 \cos (2 x)) + \sin (2 x) \cdot 1) + \frac{d}{d x} [\cos (2 x)]) - (2 x \sin (2 x) + \cos (2 x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.6.5

Multiply $\sin (2 x)$ by $1$ .

$- 8 \frac{x^{2} (2 (x (2 \cos (2 x)) + \sin (2 x)) + \frac{d}{d x} [\cos (2 x)]) - (2 x \sin (2 x) + \cos (2 x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.7

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 2.7.1

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

$- 8 \frac{x^{2} (2 (x (2 \cos (2 x)) + \sin (2 x)) + \frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d x} [2 x]) - (2 x \sin (2 x) + \cos (2 x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.7.2

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

$- 8 \frac{x^{2} (2 (x (2 \cos (2 x)) + \sin (2 x)) - \sin (u_{2}) \frac{d}{d x} [2 x]) - (2 x \sin (2 x) + \cos (2 x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.7.3

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

$- 8 \frac{x^{2} (2 (x (2 \cos (2 x)) + \sin (2 x)) - \sin (2 x) \frac{d}{d x} [2 x]) - (2 x \sin (2 x) + \cos (2 x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.8

Differentiate.

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Step 2.8.1

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

$- 8 \frac{x^{2} (2 (x (2 \cos (2 x)) + \sin (2 x)) - \sin (2 x) (2 \frac{d}{d x} [x])) - (2 x \sin (2 x) + \cos (2 x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.8.2

Multiply $2$ by $- 1$ .

$- 8 \frac{x^{2} (2 (x (2 \cos (2 x)) + \sin (2 x)) - 2 \sin (2 x) \frac{d}{d x} [x]) - (2 x \sin (2 x) + \cos (2 x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.8.3

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

$- 8 \frac{x^{2} (2 (x (2 \cos (2 x)) + \sin (2 x)) - 2 \sin (2 x) \cdot 1) - (2 x \sin (2 x) + \cos (2 x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.8.4

Multiply $- 2$ by $1$ .

$- 8 \frac{x^{2} (2 (x (2 \cos (2 x)) + \sin (2 x)) - 2 \sin (2 x)) - (2 x \sin (2 x) + \cos (2 x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.8.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$- 8 \frac{x^{2} (2 (x (2 \cos (2 x)) + \sin (2 x)) - 2 \sin (2 x)) - (2 x \sin (2 x) + \cos (2 x)) (2 x)}{x^{4}}$

Step 2.8.6

Simplify with factoring out.

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Step 2.8.6.1

Multiply $2$ by $- 1$ .

$- 8 \frac{x^{2} (2 (x (2 \cos (2 x)) + \sin (2 x)) - 2 \sin (2 x)) - 2 (2 x \sin (2 x) + \cos (2 x)) x}{x^{4}}$

Step 2.8.6.2

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

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Step 2.8.6.2.1

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

$- 8 \frac{x (x (2 (x (2 \cos (2 x)) + \sin (2 x)) - 2 \sin (2 x))) - 2 (2 x \sin (2 x) + \cos (2 x)) x}{x^{4}}$

Step 2.8.6.2.2

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

$- 8 \frac{x (x (2 (x (2 \cos (2 x)) + \sin (2 x)) - 2 \sin (2 x))) + x (- 2 (2 x \sin (2 x) + \cos (2 x)))}{x^{4}}$

Step 2.8.6.2.3

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

$- 8 \frac{x (x (2 (x (2 \cos (2 x)) + \sin (2 x)) - 2 \sin (2 x)) - 2 (2 x \sin (2 x) + \cos (2 x)))}{x^{4}}$

Step 2.9

Cancel the common factors.

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Step 2.9.1

Factor $x$ out of $x^{4}$ .

$- 8 \frac{x (x (2 (x (2 \cos (2 x)) + \sin (2 x)) - 2 \sin (2 x)) - 2 (2 x \sin (2 x) + \cos (2 x)))}{x \cdot x^{3}}$

Step 2.9.2

Cancel the common factor.

$- 8 \frac{x (x (2 (x (2 \cos (2 x)) + \sin (2 x)) - 2 \sin (2 x)) - 2 (2 x \sin (2 x) + \cos (2 x)))}{x \cdot x^{3}}$

Step 2.9.3

Rewrite the expression.

$- 8 \frac{x (2 (x (2 \cos (2 x)) + \sin (2 x)) - 2 \sin (2 x)) - 2 (2 x \sin (2 x) + \cos (2 x))}{x^{3}}$

Step 2.10

Combine $- 8$ and $\frac{x (2 (x (2 \cos (2 x)) + \sin (2 x)) - 2 \sin (2 x)) - 2 (2 x \sin (2 x) + \cos (2 x))}{x^{3}}$ .

$\frac{- 8 (x (2 (x (2 \cos (2 x)) + \sin (2 x)) - 2 \sin (2 x)) - 2 (2 x \sin (2 x) + \cos (2 x)))}{x^{3}}$

Step 2.11

Move the negative in front of the fraction.

$- \frac{8 (x (2 (x (2 \cos (2 x)) + \sin (2 x)) - 2 \sin (2 x)) - 2 (2 x \sin (2 x) + \cos (2 x)))}{x^{3}}$

Step 2.12

Simplify.

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Step 2.12.1

Apply the distributive property.

$- \frac{8 (x (2 (x (2 \cos (2 x))) + 2 \sin (2 x) - 2 \sin (2 x)) - 2 (2 x \sin (2 x) + \cos (2 x)))}{x^{3}}$

Step 2.12.2

Apply the distributive property.

$- \frac{8 (x (2 (x (2 \cos (2 x)))) + x (2 \sin (2 x)) + x (- 2 \sin (2 x)) - 2 (2 x \sin (2 x) + \cos (2 x)))}{x^{3}}$

Step 2.12.3

Apply the distributive property.

$- \frac{8 (x (2 (x (2 \cos (2 x)))) + x (2 \sin (2 x)) + x (- 2 \sin (2 x)) - 2 (2 x \sin (2 x)) - 2 \cos (2 x))}{x^{3}}$

Step 2.12.4

Apply the distributive property.

$- \frac{8 (x (2 (x (2 \cos (2 x))))) + 8 (x (2 \sin (2 x))) + 8 (x (- 2 \sin (2 x))) + 8 (- 2 (2 x \sin (2 x))) + 8 (- 2 \cos (2 x))}{x^{3}}$

Step 2.12.5

Simplify the numerator.

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Step 2.12.5.1

Combine the opposite terms in $8 (x (2 (x (2 \cos (2 x))))) + 8 (x (2 \sin (2 x))) + 8 (x (- 2 \sin (2 x))) + 8 (- 2 (2 x \sin (2 x))) + 8 (- 2 \cos (2 x))$ .

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Step 2.12.5.1.1

Reorder the factors in the terms $8 (x (2 \sin (2 x)))$ and $8 (x (- 2 \sin (2 x)))$ .

$- \frac{8 (x (2 (x (2 \cos (2 x))))) + 2 \cdot 8 x \sin (2 x) - 2 \cdot 8 x \sin (2 x) + 8 (- 2 (2 x \sin (2 x))) + 8 (- 2 \cos (2 x))}{x^{3}}$

Step 2.12.5.1.2

Subtract $2 \cdot 8 x \sin (2 x)$ from $2 \cdot 8 x \sin (2 x)$ .

$- \frac{8 (x (2 (x (2 \cos (2 x))))) + 0 + 8 (- 2 (2 x \sin (2 x))) + 8 (- 2 \cos (2 x))}{x^{3}}$

Step 2.12.5.1.3

Add $8 (x (2 (x (2 \cos (2 x)))))$ and $0$ .

$- \frac{8 (x (2 (x (2 \cos (2 x))))) + 8 (- 2 (2 x \sin (2 x))) + 8 (- 2 \cos (2 x))}{x^{3}}$

Step 2.12.5.2

Simplify each term.

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Step 2.12.5.2.1

Multiply $x$ by $x$ by adding the exponents.

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Step 2.12.5.2.1.1

Move $x$ .

$- \frac{8 (x \cdot x (2 (2 \cos (2 x)))) + 8 (- 2 (2 x \sin (2 x))) + 8 (- 2 \cos (2 x))}{x^{3}}$

Step 2.12.5.2.1.2

Multiply $x$ by $x$ .

$- \frac{8 (x^{2} (2 (2 \cos (2 x)))) + 8 (- 2 (2 x \sin (2 x))) + 8 (- 2 \cos (2 x))}{x^{3}}$

Step 2.12.5.2.2

Rewrite using the commutative property of multiplication.

$- \frac{8 (2 \cdot 2 x^{2} \cos (2 x)) + 8 (- 2 (2 x \sin (2 x))) + 8 (- 2 \cos (2 x))}{x^{3}}$

Step 2.12.5.2.3

Multiply $2$ by $2$ .

$- \frac{8 (4 x^{2} \cos (2 x)) + 8 (- 2 (2 x \sin (2 x))) + 8 (- 2 \cos (2 x))}{x^{3}}$

Step 2.12.5.2.4

Multiply $4$ by $8$ .

$- \frac{32 (x^{2} \cos (2 x)) + 8 (- 2 (2 x \sin (2 x))) + 8 (- 2 \cos (2 x))}{x^{3}}$

Step 2.12.5.2.5

Multiply $2$ by $- 2$ .

$- \frac{32 x^{2} \cos (2 x) + 8 (- 4 (x \sin (2 x))) + 8 (- 2 \cos (2 x))}{x^{3}}$

Step 2.12.5.2.6

Multiply $- 4$ by $8$ .

$- \frac{32 x^{2} \cos (2 x) - 32 (x \sin (2 x)) + 8 (- 2 \cos (2 x))}{x^{3}}$

Step 2.12.5.2.7

Multiply $- 2$ by $8$ .

$- \frac{32 x^{2} \cos (2 x) - 32 x \sin (2 x) - 16 \cos (2 x)}{x^{3}}$

Step 2.12.6

Factor $16$ out of $32 x^{2} \cos (2 x) - 32 x \sin (2 x) - 16 \cos (2 x)$ .

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Step 2.12.6.1

Factor $16$ out of $32 x^{2} \cos (2 x)$ .

$- \frac{16 (2 x^{2} \cos (2 x)) - 32 x \sin (2 x) - 16 \cos (2 x)}{x^{3}}$

Step 2.12.6.2

Factor $16$ out of $- 32 x \sin (2 x)$ .

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

Step 2.12.6.3

Factor $16$ out of $- 16 \cos (2 x)$ .

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

Step 2.12.6.4

Factor $16$ out of $16 (2 x^{2} \cos (2 x)) + 16 (- 2 x \sin (2 x))$ .

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

Step 2.12.6.5

Factor $16$ out of $16 (2 x^{2} \cos (2 x) - 2 x \sin (2 x)) + 16 (- \cos (2 x))$ .

$f'' (x) = - \frac{16 (2 x^{2} \cos (2 x) - 2 x \sin (2 x) - \cos (2 x))}{x^{3}}$

Step 3

The second derivative of $f' (x)$ with respect to $x$ is $- \frac{16 (2 x^{2} \cos (2 x) - 2 x \sin (2 x) - \cos (2 x))}{x^{3}}$ .

$- \frac{16 (2 x^{2} \cos (2 x) - 2 x \sin (2 x) - \cos (2 x))}{x^{3}}$