Calculus Examples

$y = \sin (\frac{1}{x})$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\sin (\frac{1}{x}))$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.2

Differentiate using the Power Rule.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.3

Simplify.

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Step 3.3.1

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

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

Step 3.3.2

Combine $\cos (\frac{1}{x})$ and $\frac{1}{x^{2}}$ .

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

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = - \frac{\cos (\frac{1}{x})}{x^{2}}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = - \frac{\cos (\frac{1}{x})}{x^{2}}$