Calculus Examples

$f (x) = 6 x^{4} + 24 x^{3}$

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 $6 x^{4} + 24 x^{3}$ with respect to $x$ is $\frac{d}{d x} [6 x^{4}] + \frac{d}{d x} [24 x^{3}]$ .

$f' (x) = \frac{d}{d x} (6 x^{4}) + \frac{d}{d x} (24 x^{3})$

Step 1.1.2

Evaluate $\frac{d}{d x} [6 x^{4}]$ .

Tap for more steps...

Step 1.1.2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{4}$ with respect to $x$ is $6 \frac{d}{d x} [x^{4}]$ .

$f' (x) = 6 \frac{d}{d x} (x^{4}) + \frac{d}{d x} (24 x^{3})$

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 $n = 4$ .

$f' (x) = 6 (4 x^{3}) + \frac{d}{d x} (24 x^{3})$

Step 1.1.2.3

Multiply $4$ by $6$ .

$f' (x) = 24 x^{3} + \frac{d}{d x} (24 x^{3})$

Step 1.1.3

Evaluate $\frac{d}{d x} [24 x^{3}]$ .

Tap for more steps...

Step 1.1.3.1

Since $24$ is constant with respect to $x$ , the derivative of $24 x^{3}$ with respect to $x$ is $24 \frac{d}{d x} [x^{3}]$ .

$f' (x) = 24 x^{3} + 24 \frac{d}{d x} (x^{3})$

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 $n = 3$ .

$f' (x) = 24 x^{3} + 24 (3 x^{2})$

Step 1.1.3.3

Multiply $3$ by $24$ .

$f' (x) = 24 x^{3} + 72 x^{2}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $24 x^{3} + 72 x^{2}$ .

$24 x^{3} + 72 x^{2}$

Step 2

Set the first derivative equal to $0$ then solve the equation $24 x^{3} + 72 x^{2} = 0$ .

Tap for more steps...

Step 2.1

Set the first derivative equal to $0$ .

$24 x^{3} + 72 x^{2} = 0$

Step 2.2

Factor $24 x^{2}$ out of $24 x^{3} + 72 x^{2}$ .

Tap for more steps...

Step 2.2.1

Factor $24 x^{2}$ out of $24 x^{3}$ .

$24 x^{2} (x) + 72 x^{2} = 0$

Step 2.2.2

Factor $24 x^{2}$ out of $72 x^{2}$ .

$24 x^{2} (x) + 24 x^{2} (3) = 0$

Step 2.2.3

Factor $24 x^{2}$ out of $24 x^{2} (x) + 24 x^{2} (3)$ .

$24 x^{2} (x + 3) = 0$

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 $0$ .

$x^{2} = 0$

$x + 3 = 0$

Step 2.4

Set $x^{2}$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.4.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 2.4.2

Solve $x^{2} = 0$ for $x$ .

Tap for more steps...

Step 2.4.2.1

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

$x = \pm \sqrt{0}$

Step 2.4.2.2

Simplify $\pm \sqrt{0}$ .

Tap for more steps...

Step 2.4.2.2.1

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

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

Step 2.4.2.2.2

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

$x = \pm 0$

Step 2.4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 2.5

Set $x + 3$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.5.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 2.5.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 2.6

The final solution is all the values that make $24 x^{2} (x + 3) = 0$ true.

$x = 0, - 3$

Step 3

The values which make the derivative equal to $0$ are $0, - 3$ .

$0, - 3$

Step 4

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

$(- \infty, - 3) \cup (- 3, 0) \cup (0, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - 3)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 5.1

Replace the variable $x$ with $- 4$ in the expression.

$f' (- 4) = 24 {(- 4)}^{3} + 72 {(- 4)}^{2}$

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 $- 4$ to the power of $3$ .

$f' (- 4) = 24 \cdot - 64 + 72 {(- 4)}^{2}$

Step 5.2.1.2

Multiply $24$ by $- 64$ .

$f' (- 4) = - 1536 + 72 {(- 4)}^{2}$

Step 5.2.1.3

Raise $- 4$ to the power of $2$ .

$f' (- 4) = - 1536 + 72 \cdot 16$

Step 5.2.1.4

Multiply $72$ by $16$ .

$f' (- 4) = - 1536 + 1152$

Step 5.2.2

Add $- 1536$ and $1152$ .

$f' (- 4) = - 384$

Step 5.2.3

The final answer is $- 384$ .

$- 384$

Step 5.3

At $x = - 4$ the derivative is $- 384$ . Since this is negative, the function is decreasing on $(- \infty, - 3)$ .

Decreasing on $(- \infty, - 3)$ since $f' (x) < 0$

Step 6

Substitute a value from the interval $(- 3, 0)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 6.1

Replace the variable $x$ with $- \frac{3}{2}$ in the expression.

$f' (- \frac{3}{2}) = 24 {(- \frac{3}{2})}^{3} + 72 {(- \frac{3}{2})}^{2}$

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

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 6.2.1.1.1

Apply the product rule to $- \frac{3}{2}$ .

$f' (- \frac{3}{2}) = 24 ({(- 1)}^{3} {(\frac{3}{2})}^{3}) + 72 {(- \frac{3}{2})}^{2}$

Step 6.2.1.1.2

Apply the product rule to $\frac{3}{2}$ .

$f' (- \frac{3}{2}) = 24 ({(- 1)}^{3} (\frac{3^{3}}{2^{3}})) + 72 {(- \frac{3}{2})}^{2}$

Step 6.2.1.2

Raise $- 1$ to the power of $3$ .

$f' (- \frac{3}{2}) = 24 (- \frac{3^{3}}{2^{3}}) + 72 {(- \frac{3}{2})}^{2}$

Step 6.2.1.3

Raise $3$ to the power of $3$ .

$f' (- \frac{3}{2}) = 24 (- \frac{27}{2^{3}}) + 72 {(- \frac{3}{2})}^{2}$

Step 6.2.1.4

Raise $2$ to the power of $3$ .

$f' (- \frac{3}{2}) = 24 (- \frac{27}{8}) + 72 {(- \frac{3}{2})}^{2}$

Step 6.2.1.5

Cancel the common factor of $8$ .

Tap for more steps...

Step 6.2.1.5.1

Move the leading negative in $- \frac{27}{8}$ into the numerator.

$f' (- \frac{3}{2}) = 24 (\frac{- 27}{8}) + 72 {(- \frac{3}{2})}^{2}$

Step 6.2.1.5.2

Factor $8$ out of $24$ .

$f' (- \frac{3}{2}) = 8 (3) (\frac{- 27}{8}) + 72 {(- \frac{3}{2})}^{2}$

Step 6.2.1.5.3

Cancel the common factor.

$f' (- \frac{3}{2}) = 8 \cdot (3 (\frac{- 27}{8})) + 72 {(- \frac{3}{2})}^{2}$

Step 6.2.1.5.4

Rewrite the expression.

$f' (- \frac{3}{2}) = 3 \cdot - 27 + 72 {(- \frac{3}{2})}^{2}$

Step 6.2.1.6

Multiply $3$ by $- 27$ .

$f' (- \frac{3}{2}) = - 81 + 72 {(- \frac{3}{2})}^{2}$

Step 6.2.1.7

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 6.2.1.7.1

Apply the product rule to $- \frac{3}{2}$ .

$f' (- \frac{3}{2}) = - 81 + 72 ({(- 1)}^{2} {(\frac{3}{2})}^{2})$

Step 6.2.1.7.2

Apply the product rule to $\frac{3}{2}$ .

$f' (- \frac{3}{2}) = - 81 + 72 ({(- 1)}^{2} (\frac{3^{2}}{2^{2}}))$

Step 6.2.1.8

Raise $- 1$ to the power of $2$ .

$f' (- \frac{3}{2}) = - 81 + 72 (1 (\frac{3^{2}}{2^{2}}))$

Step 6.2.1.9

Multiply $\frac{3^{2}}{2^{2}}$ by $1$ .

$f' (- \frac{3}{2}) = - 81 + 72 (\frac{3^{2}}{2^{2}})$

Step 6.2.1.10

Raise $3$ to the power of $2$ .

$f' (- \frac{3}{2}) = - 81 + 72 (\frac{9}{2^{2}})$

Step 6.2.1.11

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

$f' (- \frac{3}{2}) = - 81 + 72 (\frac{9}{4})$

Step 6.2.1.12

Cancel the common factor of $4$ .

Tap for more steps...

Step 6.2.1.12.1

Factor $4$ out of $72$ .

$f' (- \frac{3}{2}) = - 81 + 4 (18) (\frac{9}{4})$

Step 6.2.1.12.2

Cancel the common factor.

$f' (- \frac{3}{2}) = - 81 + 4 \cdot (18 (\frac{9}{4}))$

Step 6.2.1.12.3

Rewrite the expression.

$f' (- \frac{3}{2}) = - 81 + 18 \cdot 9$

Step 6.2.1.13

Multiply $18$ by $9$ .

$f' (- \frac{3}{2}) = - 81 + 162$

Step 6.2.2

Add $- 81$ and $162$ .

$f' (- \frac{3}{2}) = 81$

Step 6.2.3

The final answer is $81$ .

$81$

Step 6.3

At $x = - \frac{3}{2}$ the derivative is $81$ . Since this is positive, the function is increasing on $(- 3, 0)$ .

Increasing on $(- 3, 0)$ since $f' (x) > 0$

Step 7

Substitute a value from the interval $(0, \infty)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 7.1

Replace the variable $x$ with $1$ in the expression.

$f' (1) = 24 {(1)}^{3} + 72 {(1)}^{2}$

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

One to any power is one.

$f' (1) = 24 \cdot 1 + 72 {(1)}^{2}$

Step 7.2.1.2

Multiply $24$ by $1$ .

$f' (1) = 24 + 72 {(1)}^{2}$

Step 7.2.1.3

One to any power is one.

$f' (1) = 24 + 72 \cdot 1$

Step 7.2.1.4

Multiply $72$ by $1$ .

$f' (1) = 24 + 72$

Step 7.2.2

Add $24$ and $72$ .

$f' (1) = 96$

Step 7.2.3

The final answer is $96$ .

$96$

Step 7.3

At $x = 1$ the derivative is $96$ . Since this is positive, the function is increasing on $(0, \infty)$ .

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

Step 8

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

Increasing on: $(- 3, 0), (0, \infty)$

Decreasing on: $(- \infty, - 3)$

Step 9