Calculus Examples

$f (x) = 5 x + \frac{2}{x}$ , $x = 2$

Step 1

By the Sum Rule, the derivative of $5 x + \frac{2}{x}$ with respect to $x$ is $\frac{d}{d x} [5 x] + \frac{d}{d x} [\frac{2}{x}]$ .

$\frac{d}{d x} [5 x] + \frac{d}{d x} [\frac{2}{x}]$

Step 2

Evaluate $\frac{d}{d x} [5 x]$ .

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Step 2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$5 \frac{d}{d x} [x] + \frac{d}{d x} [\frac{2}{x}]$

Step 2.2

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

$5 \cdot 1 + \frac{d}{d x} [\frac{2}{x}]$

Step 2.3

Multiply $5$ by $1$ .

$5 + \frac{d}{d x} [\frac{2}{x}]$

Step 3

Evaluate $\frac{d}{d x} [\frac{2}{x}]$ .

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Step 3.1

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2}{x}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{1}{x}]$ .

$5 + 2 \frac{d}{d x} [\frac{1}{x}]$

Step 3.2

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

$5 + 2 \frac{d}{d x} [x^{- 1}]$

Step 3.3

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

$5 + 2 (- x^{- 2})$

Step 3.4

Multiply $- 1$ by $2$ .

$5 - 2 x^{- 2}$

Step 4

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

$5 - 2 \frac{1}{x^{2}}$

Step 5

Simplify.

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Step 5.1

Combine terms.

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Step 5.1.1

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

$5 + \frac{- 2}{x^{2}}$

Step 5.1.2

Move the negative in front of the fraction.

$5 - \frac{2}{x^{2}}$

Step 5.2

Reorder terms.

$- \frac{2}{x^{2}} + 5$

Step 6

Evaluate the derivative at $x = 2$ .

$- \frac{2}{{(2)}^{2}} + 5$

Step 7

Simplify.

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Step 7.1

Cancel the common factor of $2$ and ${(2)}^{2}$ .

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Step 7.1.1

Factor $2$ out of $2$ .

$- \frac{2 (1)}{{(2)}^{2}} + 5$

Step 7.1.2

Cancel the common factors.

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Step 7.1.2.1

Factor $2$ out of ${(2)}^{2}$ .

$- \frac{2 (1)}{2 \cdot 2} + 5$

Step 7.1.2.2

Cancel the common factor.

$- \frac{2 \cdot 1}{2 \cdot 2} + 5$

Step 7.1.2.3

Rewrite the expression.

$- \frac{1}{2} + 5$

Step 7.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{1}{2} + 5 \cdot \frac{2}{2}$

Step 7.3

Combine $5$ and $\frac{2}{2}$ .

$- \frac{1}{2} + \frac{5 \cdot 2}{2}$

Step 7.4

Combine the numerators over the common denominator.

$\frac{- 1 + 5 \cdot 2}{2}$

Step 7.5

Simplify the numerator.

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Step 7.5.1

Multiply $5$ by $2$ .

$\frac{- 1 + 10}{2}$

Step 7.5.2

Add $- 1$ and $10$ .

$\frac{9}{2}$