Calculus Examples

$f (x) = x^{2} - 8 x$

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.

Tap for more steps...

Step 1.1.1.1

By the Sum Rule, the derivative of $x^{2} - 8 x$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 x]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 x]$

Step 1.1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x + \frac{d}{d x} [- 8 x]$

Step 1.1.2

Evaluate $\frac{d}{d x} [- 8 x]$ .

Tap for more steps...

Step 1.1.2.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x$ with respect to $x$ is $- 8 \frac{d}{d x} [x]$ .

$2 x - 8 \frac{d}{d x} [x]$

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 $n = 1$ .

$2 x - 8 \cdot 1$

Step 1.1.2.3

Multiply $- 8$ by $1$ .

$f' (x) = 2 x - 8$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $2 x - 8$ .

$2 x - 8$

Step 2

Set the first derivative equal to $0$ then solve the equation $2 x - 8 = 0$ .

Tap for more steps...

Step 2.1

Set the first derivative equal to $0$ .

$2 x - 8 = 0$

Step 2.2

Add $8$ to both sides of the equation.

$2 x = 8$

Step 2.3

Divide each term in $2 x = 8$ by $2$ and simplify.

Tap for more steps...

Step 2.3.1

Divide each term in $2 x = 8$ by $2$ .

$\frac{2 x}{2} = \frac{8}{2}$

Step 2.3.2

Simplify the left side.

Tap for more steps...

Step 2.3.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{8}{2}$

Step 2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{8}{2}$

Step 2.3.3

Simplify the right side.

Tap for more steps...

Step 2.3.3.1

Divide $8$ by $2$ .

$x = 4$

Step 3

The values which make the derivative equal to $0$ are $4$ .

$4$

Step 4

After finding the point that makes the derivative $f' (x) = 2 x - 8$ equal to $0$ or undefined, the interval to check where $f (x) = x^{2} - 8 x$ is increasing and where it is decreasing is $(- \infty, 4) \cup (4, \infty)$ .

$(- \infty, 4) \cup (4, \infty)$

Step 5

Substitute a value from the interval $(- \infty, 4)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 5.1

Replace the variable $x$ with $3$ in the expression.

$f' (3) = 2 (3) - 8$

Step 5.2

Simplify the result.

Tap for more steps...

Step 5.2.1

Multiply $2$ by $3$ .

$f' (3) = 6 - 8$

Step 5.2.2

Subtract $8$ from $6$ .

$f' (3) = - 2$

Step 5.2.3

The final answer is $- 2$ .

$- 2$

Step 5.3

At $x = 3$ the derivative is $- 2$ . Since this is negative, the function is decreasing on $(- \infty, 4)$ .

Decreasing on $(- \infty, 4)$ since $f' (x) < 0$

Step 6

Substitute a value from the interval $(4, \infty)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 6.1

Replace the variable $x$ with $5$ in the expression.

$f' (5) = 2 (5) - 8$

Step 6.2

Simplify the result.

Tap for more steps...

Step 6.2.1

Multiply $2$ by $5$ .

$f' (5) = 10 - 8$

Step 6.2.2

Subtract $8$ from $10$ .

$f' (5) = 2$

Step 6.2.3

The final answer is $2$ .

$2$

Step 6.3

At $x = 5$ the derivative is $2$ . Since this is positive, the function is increasing on $(4, \infty)$ .

Increasing on $(4, \infty)$ since $f' (x) > 0$

Step 7

List the intervals on which the function is increasing and decreasing.

Increasing on: $(4, \infty)$

Decreasing on: $(- \infty, 4)$

Step 8