Calculus Examples

$f (x) = \frac{\sin (x)}{x} + \frac{| \sin (x) |}{x}$

Step 1

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

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

Step 2

Evaluate $\frac{d}{d x} [\frac{\sin (x)}{x}]$ .

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Step 2.1

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 $g (x) = x$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Multiply $- 1$ by $1$ .

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

Step 3

Evaluate $\frac{d}{d x} [\frac{| \sin (x) |}{x}]$ .

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Step 3.1

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

Step 3.2

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

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Step 3.2.1

To apply the Chain Rule, set $u$ as $\sin (x)$ .

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

Step 3.2.2

The derivative of $| u |$ with respect to $u$ is $\frac{u}{| u |}$ .

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

Step 3.2.3

Replace all occurrences of $u$ with $\sin (x)$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Combine $\frac{\sin (x)}{| \sin (x) |}$ and $\cos (x)$ .

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

Step 3.6

Combine $x$ and $\frac{\sin (x) \cos (x)}{| \sin (x) |}$ .

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

Step 3.7

Multiply $- 1$ by $1$ .

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

Step 3.8

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

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

Step 3.9

Combine.

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

Step 3.10

Apply the distributive property.

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

Step 3.11

Cancel the common factor of $| \sin (x) |$ .

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Step 3.11.1

Cancel the common factor.

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

Step 3.11.2

Rewrite the expression.

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

Step 3.12

To multiply absolute values, multiply the terms inside each absolute value.

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

Step 3.13

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

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

Step 3.14

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

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

Step 3.15

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

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

Step 3.16

Add $1$ and $1$ .

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

Step 4

Simplify.

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Step 4.1

Combine terms.

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Step 4.1.1

To write $\frac{x \cos (x) - \sin (x)}{x^{2}}$ as a fraction with a common denominator, multiply by $\frac{| \sin (x) |}{| \sin (x) |}$ .

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

Step 4.1.2

Write each expression with a common denominator of $| \sin (x) | x^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.2.1

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

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

Step 4.1.2.2

Reorder the factors of $| \sin (x) | x^{2}$ .

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

Step 4.1.3

Combine the numerators over the common denominator.

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

Step 4.2

Reorder terms.

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

Step 4.3

Simplify the numerator.

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Step 4.3.1

Apply the distributive property.

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

Step 4.3.2

Rewrite using the commutative property of multiplication.

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

Step 4.3.3

Remove non-negative terms from the absolute value.

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

Step 4.3.4

Rewrite $| \sin (x) | x \cos (x) - | \sin (x) | \sin (x) + x \cos (x) \sin (x) - \sin^{2} (x)$ in a factored form.

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Step 4.3.4.1

Reorder terms.

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

Step 4.3.4.2

Factor out the greatest common factor from each group.

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Step 4.3.4.2.1

Group the first two terms and the last two terms.

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

Step 4.3.4.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 4.3.4.3

Factor the polynomial by factoring out the greatest common factor, $x \cos (x) - \sin (x)$ .

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