Calculus Examples

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

Step 1

Find the derivative.

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Since $3$ is constant with respect to $x$ , the derivative of $\frac{3}{x}$ with respect to $x$ is $3 \frac{d}{d x} [\frac{1}{x}]$ .

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

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

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

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

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

Multiply $- 1$ by $3$ .

$- 3 x^{- 2}$

Simplify.

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

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

Combine terms.

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

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

Move the negative in front of the fraction.

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

Step 2

Set the derivative equal to $0$ then solve the equation $- \frac{3}{x^{2}} = 0$ .

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Set the numerator equal to zero.

$3 = 0$

Since $3 \neq 0$ , there are no solutions.

No solution

Step 3

There are no solution found by setting the derivative equal to $0$ , $- \frac{3}{x^{2}} = 0$ so there are no horizontal tangent lines.

No horizontal tangent lines found

Step 4