Calculus Examples

$f (x) = x + \frac{3}{x}$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $x + \frac{3}{x}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [\frac{3}{x}]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [\frac{3}{x}]$

Step 1.2

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

$1 + \frac{d}{d x} [\frac{3}{x}]$

Step 2

Evaluate $\frac{d}{d x} [\frac{3}{x}]$ .

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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 $3 \frac{d}{d x} [\frac{1}{x}]$ .

$1 + 3 \frac{d}{d x} [\frac{1}{x}]$

Step 2.2

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

$1 + 3 \frac{d}{d x} [x^{- 1}]$

Step 2.3

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

$1 + 3 (- x^{- 2})$

Step 2.4

Multiply $- 1$ by $3$ .

$1 - 3 x^{- 2}$

Step 3

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

$1 - 3 \frac{1}{x^{2}}$

Step 4

Simplify.

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Step 4.1

Combine terms.

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Step 4.1.1

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

$1 + \frac{- 3}{x^{2}}$

Step 4.1.2

Move the negative in front of the fraction.

$1 - \frac{3}{x^{2}}$

Step 4.2

Reorder terms.

$- \frac{3}{x^{2}} + 1$