Calculus Examples

$y = \sin (\frac{3}{x})$

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{3}{x}$ .

$\frac{d}{d u} [\sin (u)] \frac{d}{d x} [\frac{3}{x}]$

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $\frac{3}{x}$ .

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

Step 2

Differentiate.

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Step 2.1

Since $3$ is constant with respect to $x$ , the derivative of $\frac{3}{x}$ with respect to $x$ is $3 \frac{d}{d x} [\frac{1}{x}]$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Multiply $- 1$ by $3$ .

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

Step 3

Simplify.

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Step 3.1

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

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

Step 3.2

Combine terms.

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Step 3.2.1

Combine $- 3$ and $\frac{1}{x^{2}}$ .

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

Step 3.2.2

Move the negative in front of the fraction.

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

Step 3.2.3

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

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

Step 3.2.4

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

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