Calculus Examples

$y = x \sin (\frac{5}{x})$

Step 1

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 $g (x) = \sin (\frac{5}{x})$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = \frac{5}{x}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $\frac{5}{x}$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $\frac{5}{x}$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Multiply $- 1$ by $5$ .

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

Step 4

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

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

Step 5

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

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

Step 6

Simplify the expression.

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Step 6.1

Add $- 2$ and $1$ .

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

Step 6.2

Move $- 5$ to the left of $\cos (\frac{5}{x})$ .

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

Step 7

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

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

Step 8

Multiply $\sin (\frac{5}{x})$ by $1$ .

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

Step 9

Simplify.

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Step 9.1

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

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

Step 9.2

Combine terms.

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Step 9.2.1

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

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

Step 9.2.2

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

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

Step 9.2.3

Move the negative in front of the fraction.

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