Calculus Examples

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

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x - 1$ and $g (x) = x$ .

$\frac{x \frac{d}{d x} [x - 1] - (x - 1) \frac{d}{d x} [x]}{x^{2}}$

Step 1.1.2

Differentiate.

Tap for more steps...

Step 1.1.2.1

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

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

Step 1.1.2.2

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

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

Step 1.1.2.3

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$\frac{x (1 + 0) - (x - 1) \frac{d}{d x} [x]}{x^{2}}$

Step 1.1.2.4

Simplify the expression.

Tap for more steps...

Step 1.1.2.4.1

Add $1$ and $0$ .

$\frac{x \cdot 1 - (x - 1) \frac{d}{d x} [x]}{x^{2}}$

Step 1.1.2.4.2

Multiply $x$ by $1$ .

$\frac{x - (x - 1) \frac{d}{d x} [x]}{x^{2}}$

Step 1.1.2.5

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

$\frac{x - (x - 1) \cdot 1}{x^{2}}$

Step 1.1.2.6

Multiply $- 1$ by $1$ .

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

Step 1.1.3

Simplify.

Tap for more steps...

Step 1.1.3.1

Apply the distributive property.

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

Step 1.1.3.2

Simplify the numerator.

Tap for more steps...

Step 1.1.3.2.1

Subtract $x$ from $x$ .

$\frac{0 - - 1}{x^{2}}$

Step 1.1.3.2.2

Subtract $- 1$ from $0$ .

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

Step 1.1.3.2.3

Multiply $- 1$ by $- 1$ .

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

Step 1.2

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

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

Step 2

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

Tap for more steps...

Step 2.1

Set the first derivative equal to $0$ .

$\frac{1}{x^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

$1 = 0$

Step 2.3

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

No solution

Step 3

Find the values where the derivative is undefined.

Tap for more steps...

Step 3.1

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

$x^{2} = 0$

Step 3.2

Solve for $x$ .

Tap for more steps...

Step 3.2.1

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

$x = \pm \sqrt{0}$

Step 3.2.2

Simplify $\pm \sqrt{0}$ .

Tap for more steps...

Step 3.2.2.1

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

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

Step 3.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 3.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 4

Evaluate $\frac{x - 1}{x}$ at each $x$ value where the derivative is $0$ or undefined.

Tap for more steps...

Step 4.1

Evaluate at $x = 0$ .

Tap for more steps...

Step 4.1.1

Substitute $0$ for $x$ .

$\frac{(0) - 1}{0}$

Step 4.1.2

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 5

There are no values of $x$ in the domain of the original problem where the derivative is $0$ or undefined.

No critical points found