Calculus Examples

$f (x) = 3 x^{3} + 18 x^{2} - 45 x$

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

By the Sum Rule, the derivative of $3 x^{3} + 18 x^{2} - 45 x$ with respect to $x$ is $\frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [18 x^{2}] + \frac{d}{d x} [- 45 x]$ .

$f' (x) = \frac{d}{d x} (3 x^{3}) + \frac{d}{d x} (18 x^{2}) + \frac{d}{d x} (- 45 x)$

Step 1.1.2

Evaluate $\frac{d}{d x} [3 x^{3}]$ .

Tap for more steps...

Step 1.1.2.1

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

$f' (x) = 3 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (18 x^{2}) + \frac{d}{d x} (- 45 x)$

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 $n = 3$ .

$f' (x) = 3 (3 x^{2}) + \frac{d}{d x} (18 x^{2}) + \frac{d}{d x} (- 45 x)$

Step 1.1.2.3

Multiply $3$ by $3$ .

$f' (x) = 9 x^{2} + \frac{d}{d x} (18 x^{2}) + \frac{d}{d x} (- 45 x)$

Step 1.1.3

Evaluate $\frac{d}{d x} [18 x^{2}]$ .

Tap for more steps...

Step 1.1.3.1

Since $18$ is constant with respect to $x$ , the derivative of $18 x^{2}$ with respect to $x$ is $18 \frac{d}{d x} [x^{2}]$ .

$f' (x) = 9 x^{2} + 18 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 45 x)$

Step 1.1.3.2

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

$f' (x) = 9 x^{2} + 18 (2 x) + \frac{d}{d x} (- 45 x)$

Step 1.1.3.3

Multiply $2$ by $18$ .

$f' (x) = 9 x^{2} + 36 x + \frac{d}{d x} (- 45 x)$

Step 1.1.4

Evaluate $\frac{d}{d x} [- 45 x]$ .

Tap for more steps...

Step 1.1.4.1

Since $- 45$ is constant with respect to $x$ , the derivative of $- 45 x$ with respect to $x$ is $- 45 \frac{d}{d x} [x]$ .

$f' (x) = 9 x^{2} + 36 x - 45 \frac{d}{d x} x$

Step 1.1.4.2

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

$f' (x) = 9 x^{2} + 36 x - 45 \cdot 1$

Step 1.1.4.3

Multiply $- 45$ by $1$ .

$f' (x) = 9 x^{2} + 36 x - 45$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $9 x^{2} + 36 x - 45$ .

$9 x^{2} + 36 x - 45$

Step 2

Set the first derivative equal to $0$ then solve the equation $9 x^{2} + 36 x - 45 = 0$ .

Tap for more steps...

Step 2.1

Set the first derivative equal to $0$ .

$9 x^{2} + 36 x - 45 = 0$

Step 2.2

Factor the left side of the equation.

Tap for more steps...

Step 2.2.1

Factor $9$ out of $9 x^{2} + 36 x - 45$ .

Tap for more steps...

Step 2.2.1.1

Factor $9$ out of $9 x^{2}$ .

$9 (x^{2}) + 36 x - 45 = 0$

Step 2.2.1.2

Factor $9$ out of $36 x$ .

$9 (x^{2}) + 9 (4 x) - 45 = 0$

Step 2.2.1.3

Factor $9$ out of $- 45$ .

$9 x^{2} + 9 (4 x) + 9 \cdot - 5 = 0$

Step 2.2.1.4

Factor $9$ out of $9 x^{2} + 9 (4 x)$ .

$9 (x^{2} + 4 x) + 9 \cdot - 5 = 0$

Step 2.2.1.5

Factor $9$ out of $9 (x^{2} + 4 x) + 9 \cdot - 5$ .

$9 (x^{2} + 4 x - 5) = 0$

Step 2.2.2

Factor.

Tap for more steps...

Step 2.2.2.1

Factor $x^{2} + 4 x - 5$ using the AC method.

Tap for more steps...

Step 2.2.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 5$ and whose sum is $4$ .

$- 1, 5$

Step 2.2.2.1.2

Write the factored form using these integers.

$9 ((x - 1) (x + 5)) = 0$

Step 2.2.2.2

Remove unnecessary parentheses.

$9 (x - 1) (x + 5) = 0$

Step 2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 1 = 0$

$x + 5 = 0$

Step 2.4

Set $x - 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.4.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 2.4.2

Add $1$ to both sides of the equation.

$x = 1$

Step 2.5

Set $x + 5$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.5.1

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Step 2.5.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 2.6

The final solution is all the values that make $9 (x - 1) (x + 5) = 0$ true.

$x = 1, - 5$

Step 3

The values which make the derivative equal to $0$ are $1, - 5$ .

$1, - 5$

Step 4

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the derivative $0$ or undefined.

$(- \infty, - 5) \cup (- 5, 1) \cup (1, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - 5)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 5.1

Replace the variable $x$ with $- 6$ in the expression.

$f' (- 6) = 9 {(- 6)}^{2} + 36 (- 6) - 45$

Step 5.2

Simplify the result.

Tap for more steps...

Step 5.2.1

Simplify each term.

Tap for more steps...

Step 5.2.1.1

Raise $- 6$ to the power of $2$ .

$f' (- 6) = 9 \cdot 36 + 36 (- 6) - 45$

Step 5.2.1.2

Multiply $9$ by $36$ .

$f' (- 6) = 324 + 36 (- 6) - 45$

Step 5.2.1.3

Multiply $36$ by $- 6$ .

$f' (- 6) = 324 - 216 - 45$

Step 5.2.2

Simplify by subtracting numbers.

Tap for more steps...

Step 5.2.2.1

Subtract $216$ from $324$ .

$f' (- 6) = 108 - 45$

Step 5.2.2.2

Subtract $45$ from $108$ .

$f' (- 6) = 63$

Step 5.2.3

The final answer is $63$ .

$63$

Step 5.3

At $x = - 6$ the derivative is $63$ . Since this is positive, the function is increasing on $(- \infty, - 5)$ .

Increasing on $(- \infty, - 5)$ since $f' (x) > 0$

Step 6

Substitute a value from the interval $(- 5, 1)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 6.1

Replace the variable $x$ with $- 2$ in the expression.

$f' (- 2) = 9 {(- 2)}^{2} + 36 (- 2) - 45$

Step 6.2

Simplify the result.

Tap for more steps...

Step 6.2.1

Simplify each term.

Tap for more steps...

Step 6.2.1.1

Raise $- 2$ to the power of $2$ .

$f' (- 2) = 9 \cdot 4 + 36 (- 2) - 45$

Step 6.2.1.2

Multiply $9$ by $4$ .

$f' (- 2) = 36 + 36 (- 2) - 45$

Step 6.2.1.3

Multiply $36$ by $- 2$ .

$f' (- 2) = 36 - 72 - 45$

Step 6.2.2

Simplify by subtracting numbers.

Tap for more steps...

Step 6.2.2.1

Subtract $72$ from $36$ .

$f' (- 2) = - 36 - 45$

Step 6.2.2.2

Subtract $45$ from $- 36$ .

$f' (- 2) = - 81$

Step 6.2.3

The final answer is $- 81$ .

$- 81$

Step 6.3

At $x = - 2$ the derivative is $- 81$ . Since this is negative, the function is decreasing on $(- 5, 1)$ .

Decreasing on $(- 5, 1)$ since $f' (x) < 0$

Step 7

Substitute a value from the interval $(1, \infty)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 7.1

Replace the variable $x$ with $2$ in the expression.

$f' (2) = 9 {(2)}^{2} + 36 (2) - 45$

Step 7.2

Simplify the result.

Tap for more steps...

Step 7.2.1

Simplify each term.

Tap for more steps...

Step 7.2.1.1

Raise $2$ to the power of $2$ .

$f' (2) = 9 \cdot 4 + 36 (2) - 45$

Step 7.2.1.2

Multiply $9$ by $4$ .

$f' (2) = 36 + 36 (2) - 45$

Step 7.2.1.3

Multiply $36$ by $2$ .

$f' (2) = 36 + 72 - 45$

Step 7.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 7.2.2.1

Add $36$ and $72$ .

$f' (2) = 108 - 45$

Step 7.2.2.2

Subtract $45$ from $108$ .

$f' (2) = 63$

Step 7.2.3

The final answer is $63$ .

$63$

Step 7.3

At $x = 2$ the derivative is $63$ . Since this is positive, the function is increasing on $(1, \infty)$ .

Increasing on $(1, \infty)$ since $f' (x) > 0$

Step 8

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

Increasing on: $(- \infty, - 5), (1, \infty)$

Decreasing on: $(- 5, 1)$

Step 9