Calculus Examples

$f (x) = 5 x \sin (\frac{1}{x})$

Step 1

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

$5 \frac{d}{d x} [x \sin (\frac{1}{x})]$

Step 2

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 $g (x) = \sin (\frac{1}{x})$ .

$5 (x \frac{d}{d x} [\sin (\frac{1}{x})] + \sin (\frac{1}{x}) \frac{d}{d x} [x])$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = \frac{1}{x}$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $\frac{1}{x}$ .

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

Step 3.2

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

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

Step 3.3

Replace all occurrences of $u$ with $\frac{1}{x}$ .

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

Step 4

Differentiate using the Power Rule.

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Step 4.1

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

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

Step 4.2

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

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

Step 5

Raise $x$ to the power of $1$ .

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

Step 6

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

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

Step 7

Simplify the expression.

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Step 7.1

Add $- 2$ and $1$ .

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

Step 7.2

Move $- 1$ to the left of $\cos (\frac{1}{x})$ .

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

Step 7.3

Rewrite $- 1 \cos (\frac{1}{x})$ as $- \cos (\frac{1}{x})$ .

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

Step 8

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

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

Step 9

Multiply $\sin (\frac{1}{x})$ by $1$ .

$5 (x^{- 1} (- \cos (\frac{1}{x})) + \sin (\frac{1}{x}))$

Step 10

Simplify.

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Step 10.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$5 (\frac{1}{x} (- \cos (\frac{1}{x})) + \sin (\frac{1}{x}))$

Step 10.2

Apply the distributive property.

$5 (\frac{1}{x} (- \cos (\frac{1}{x}))) + 5 \sin (\frac{1}{x})$

Step 10.3

Combine terms.

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Step 10.3.1

Combine $\frac{1}{x}$ and $\cos (\frac{1}{x})$ .

$5 (- \frac{\cos (\frac{1}{x})}{x}) + 5 \sin (\frac{1}{x})$

Step 10.3.2

Multiply $- 1$ by $5$ .

$- 5 \frac{\cos (\frac{1}{x})}{x} + 5 \sin (\frac{1}{x})$

Step 10.3.3

Combine $- 5$ and $\frac{\cos (\frac{1}{x})}{x}$ .

$\frac{- 5 \cos (\frac{1}{x})}{x} + 5 \sin (\frac{1}{x})$

Step 10.3.4

Move the negative in front of the fraction.

$- \frac{5 \cos (\frac{1}{x})}{x} + 5 \sin (\frac{1}{x})$