Calculus Examples

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

Step 1

Set $y$ as a function of $x$ .

$f (x) = \frac{1}{x}$

Step 2

Find the derivative.

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Step 2.1

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

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

Step 2.2

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

$- x^{- 2}$

Step 2.3

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

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

Step 3

Set the derivative equal to $0$ then solve the equation $- \frac{1}{x^{2}} = 0$ .

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Step 3.1

Set the numerator equal to zero.

$1 = 0$

Step 3.2

Since $1 \neq 0$ , there are no solutions.

No solution

Step 4

There are no solution found by setting the derivative equal to $0$ , $- \frac{1}{x^{2}} = 0$ so there are no horizontal tangent lines.

No horizontal tangent lines found

Step 5