Calculus Examples

$y = x + \frac{1}{x}$

Step 1

Differentiate both sides of the equation.

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

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.

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Step 3.1.1

By the Sum Rule, the derivative of $x + \frac{1}{x}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [\frac{1}{x}]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [\frac{1}{x}]$

Step 3.1.2

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

$1 + \frac{d}{d x} [\frac{1}{x}]$

Step 3.2

Evaluate $\frac{d}{d x} [\frac{1}{x}]$ .

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Step 3.2.1

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

$1 + \frac{d}{d x} [x^{- 1}]$

Step 3.2.2

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

$1 - x^{- 2}$

Step 3.3

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

$1 - \frac{1}{x^{2}}$

Step 3.4

Reorder terms.

$- \frac{1}{x^{2}} + 1$

Step 4

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

$y' = - \frac{1}{x^{2}} + 1$

Step 5

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

$\frac{d y}{d x} = - \frac{1}{x^{2}} + 1$