Calculus Examples

$y = x^{2} \sec (\frac{1}{x})$

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 $f (x) = x^{2}$ and $g (x) = \sec (\frac{1}{x})$ .

$x^{2} \frac{d}{d x} [\sec (\frac{1}{x})] + \sec (\frac{1}{x}) \frac{d}{d x} [x^{2}]$

Step 2

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

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Step 2.1

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

$x^{2} (\frac{d}{d u} [\sec (u)] \frac{d}{d x} [\frac{1}{x}]) + \sec (\frac{1}{x}) \frac{d}{d x} [x^{2}]$

Step 2.2

The derivative of $\sec (u)$ with respect to $u$ is $\sec (u) \tan (u)$ .

$x^{2} (\sec (u) \tan (u) \frac{d}{d x} [\frac{1}{x}]) + \sec (\frac{1}{x}) \frac{d}{d x} [x^{2}]$

Step 2.3

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

$x^{2} (\sec (\frac{1}{x}) \tan (\frac{1}{x}) \frac{d}{d x} [\frac{1}{x}]) + \sec (\frac{1}{x}) \frac{d}{d x} [x^{2}]$

Step 3

Differentiate using the Power Rule.

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Step 3.1

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

$x^{2} (\sec (\frac{1}{x}) \tan (\frac{1}{x}) \frac{d}{d x} [x^{- 1}]) + \sec (\frac{1}{x}) \frac{d}{d x} [x^{2}]$

Step 3.2

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

$x^{2} (\sec (\frac{1}{x}) \tan (\frac{1}{x}) (- x^{- 2})) + \sec (\frac{1}{x}) \frac{d}{d x} [x^{2}]$

Step 4

Multiply $x^{2}$ by $x^{- 2}$ by adding the exponents.

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Step 4.1

Move $x^{- 2}$ .

$x^{- 2} x^{2} (\sec (\frac{1}{x}) \tan (\frac{1}{x}) \cdot (- 1)) + \sec (\frac{1}{x}) \frac{d}{d x} [x^{2}]$

Step 4.2

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

$x^{- 2 + 2} (\sec (\frac{1}{x}) \tan (\frac{1}{x}) \cdot (- 1)) + \sec (\frac{1}{x}) \frac{d}{d x} [x^{2}]$

Step 4.3

Add $- 2$ and $2$ .

$x^{0} (\sec (\frac{1}{x}) \tan (\frac{1}{x}) \cdot (- 1)) + \sec (\frac{1}{x}) \frac{d}{d x} [x^{2}]$

Step 5

Simplify the expression.

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Step 5.1

Simplify $x^{0} (\sec (\frac{1}{x}) \tan (\frac{1}{x}) \cdot (- 1))$ .

$\sec (\frac{1}{x}) \tan (\frac{1}{x}) \cdot (- 1) + \sec (\frac{1}{x}) \frac{d}{d x} [x^{2}]$

Step 5.2

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

$- 1 \cdot (\sec (\frac{1}{x}) \tan (\frac{1}{x})) + \sec (\frac{1}{x}) \frac{d}{d x} [x^{2}]$

Step 5.3

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

$- \sec (\frac{1}{x}) \tan (\frac{1}{x}) + \sec (\frac{1}{x}) \frac{d}{d x} [x^{2}]$

Step 6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$- \sec (\frac{1}{x}) \tan (\frac{1}{x}) + \sec (\frac{1}{x}) (2 x)$

Step 7

Reorder terms.

$- \sec (\frac{1}{x}) \tan (\frac{1}{x}) + 2 x \sec (\frac{1}{x})$