Calculus Examples

$f (x) = \frac{1}{x}$ , $[- 6, 6]$

Find the derivative.

Tap for more steps...

Find the first derivative.

Tap for more steps...

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

$f' (x) = \frac{d}{d x} (x^{- 1})$

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

$f' (x) = - x^{- 2}$

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

$f' (x) = - \frac{1}{x^{2}}$

The first derivative of $f (x)$ with respect to $x$ is $- \frac{1}{x^{2}}$ .

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

Find if the derivative is continuous on $[- 6, 6]$ .

Tap for more steps...

To find whether the function is continuous on $[- 6, 6]$ or not, find the domain of $f' (x) = - \frac{1}{x^{2}}$ .

Tap for more steps...

Set the denominator in $\frac{1}{x^{2}}$ equal to $0$ to find where the expression is undefined.

$x^{2} = 0$

Solve for $x$ .

Tap for more steps...

Take the $square$ root of both sides of the $equation$ to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Simplify the right side of the equation.

Tap for more steps...

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Pull terms out from under the radical.

$x = \pm | 0 |$

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

$x = \pm 0$

$\pm 0$ is equal to $0$ .

$x = 0$

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

$f' (x)$ is not continuous on $[- 6, 6]$ because $0$ is not in the domain of $f' (x) = - \frac{1}{x^{2}}$ .

The function is not continuous.

The function is not differentiable on $[- 6, 6]$ because the derivative $- \frac{1}{x^{2}}$ is not continuous on $[- 6, 6]$ .

The function is not differentiable.

Enter YOUR Problem