Calculus Examples

$f (x) = 2 x^{4} + 6 x^{3}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

By the Sum Rule, the derivative of $2 x^{4} + 6 x^{3}$ with respect to $x$ is $\frac{d}{d x} [2 x^{4}] + \frac{d}{d x} [6 x^{3}]$ .

$f' (x) = \frac{d}{d x} (2 x^{4}) + \frac{d}{d x} (6 x^{3})$

Step 1.1.2

Evaluate $\frac{d}{d x} [2 x^{4}]$ .

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Step 1.1.2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{4}$ with respect to $x$ is $2 \frac{d}{d x} [x^{4}]$ .

$f' (x) = 2 \frac{d}{d x} (x^{4}) + \frac{d}{d x} (6 x^{3})$

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 $n = 4$ .

$f' (x) = 2 (4 x^{3}) + \frac{d}{d x} (6 x^{3})$

Step 1.1.2.3

Multiply $4$ by $2$ .

$f' (x) = 8 x^{3} + \frac{d}{d x} (6 x^{3})$

Step 1.1.3

Evaluate $\frac{d}{d x} [6 x^{3}]$ .

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Step 1.1.3.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{3}$ with respect to $x$ is $6 \frac{d}{d x} [x^{3}]$ .

$f' (x) = 8 x^{3} + 6 \frac{d}{d x} (x^{3})$

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 $n = 3$ .

$f' (x) = 8 x^{3} + 6 (3 x^{2})$

Step 1.1.3.3

Multiply $3$ by $6$ .

$f' (x) = 8 x^{3} + 18 x^{2}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $8 x^{3} + 18 x^{2}$ .

$8 x^{3} + 18 x^{2}$

Step 2

Set the first derivative equal to $0$ then solve the equation $8 x^{3} + 18 x^{2} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$8 x^{3} + 18 x^{2} = 0$

Step 2.2

Factor $2 x^{2}$ out of $8 x^{3} + 18 x^{2}$ .

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Step 2.2.1

Factor $2 x^{2}$ out of $8 x^{3}$ .

$2 x^{2} (4 x) + 18 x^{2} = 0$

Step 2.2.2

Factor $2 x^{2}$ out of $18 x^{2}$ .

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

Step 2.2.3

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

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

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 $0$ .

$x^{2} = 0$

$4 x + 9 = 0$

Step 2.4

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

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Step 2.4.1

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

$x^{2} = 0$

Step 2.4.2

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

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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 $\pm \sqrt{0}$ .

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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 $4 x + 9$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $4 x + 9$ equal to $0$ .

$4 x + 9 = 0$

Step 2.5.2

Solve $4 x + 9 = 0$ for $x$ .

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Step 2.5.2.1

Subtract $9$ from both sides of the equation.

$4 x = - 9$

Step 2.5.2.2

Divide each term in $4 x = - 9$ by $4$ and simplify.

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Step 2.5.2.2.1

Divide each term in $4 x = - 9$ by $4$ .

$\frac{4 x}{4} = \frac{- 9}{4}$

Step 2.5.2.2.2

Simplify the left side.

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Step 2.5.2.2.2.1

Cancel the common factor of $4$ .

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Step 2.5.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 9}{4}$

Step 2.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 9}{4}$

Step 2.5.2.2.3

Simplify the right side.

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Step 2.5.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{9}{4}$

Step 2.6

The final solution is all the values that make $2 x^{2} (4 x + 9) = 0$ true.

$x = 0, - \frac{9}{4}$

Step 3

The values which make the derivative equal to $0$ are $0, - \frac{9}{4}$ .

$0, - \frac{9}{4}$

Step 4

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

$(- \infty, - \frac{9}{4}) \cup (- \frac{9}{4}, 0) \cup (0, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - \frac{9}{4})$ into the derivative to determine if the function is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 3.25$ in the expression.

$f' (- 3.25) = 8 {(- 3.25)}^{3} + 18 {(- 3.25)}^{2}$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Raise $- 3.25$ to the power of $3$ .

$f' (- 3.25) = 8 \cdot - 34.328125 + 18 {(- 3.25)}^{2}$

Step 5.2.1.2

Multiply $8$ by $- 34.328125$ .

$f' (- 3.25) = - 274.625 + 18 {(- 3.25)}^{2}$

Step 5.2.1.3

Raise $- 3.25$ to the power of $2$ .

$f' (- 3.25) = - 274.625 + 18 \cdot 10.5625$

Step 5.2.1.4

Multiply $18$ by $10.5625$ .

$f' (- 3.25) = - 274.625 + 190.125$

Step 5.2.2

Add $- 274.625$ and $190.125$ .

$f' (- 3.25) = - 84.5$

Step 5.2.3

The final answer is $- 84.5$ .

$- 84.5$

Step 5.3

At $x = - 3.25$ the derivative is $- 84.5$ . Since this is negative, the function is decreasing on $(- \infty, - \frac{9}{4})$ .

Decreasing on $(- \infty, - \frac{9}{4})$ since $f' (x) < 0$

Step 6

Substitute a value from the interval $(- 2.25, 0)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 1.125$ in the expression.

$f' (- 1.125) = 8 {(- 1.125)}^{3} + 18 {(- 1.125)}^{2}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Raise $- 1.125$ to the power of $3$ .

$f' (- 1.125) = 8 \cdot - 1.42382812 + 18 {(- 1.125)}^{2}$

Step 6.2.1.2

Multiply $8$ by $- 1.42382812$ .

$f' (- 1.125) = - 11.390625 + 18 {(- 1.125)}^{2}$

Step 6.2.1.3

Raise $- 1.125$ to the power of $2$ .

$f' (- 1.125) = - 11.390625 + 18 \cdot 1.265625$

Step 6.2.1.4

Multiply $18$ by $1.265625$ .

$f' (- 1.125) = - 11.390625 + 22.78125$

Step 6.2.2

Add $- 11.390625$ and $22.78125$ .

$f' (- 1.125) = 11.390625$

Step 6.2.3

The final answer is $11.390625$ .

$11.390625$

Step 6.3

At $x = - 1.125$ the derivative is $11.390625$ . Since this is positive, the function is increasing on $(- 2.25, 0)$ .

Increasing on $(- \frac{9}{4}, 0)$ 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.

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Step 7.1

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

$f' (1) = 8 {(1)}^{3} + 18 {(1)}^{2}$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

One to any power is one.

$f' (1) = 8 \cdot 1 + 18 {(1)}^{2}$

Step 7.2.1.2

Multiply $8$ by $1$ .

$f' (1) = 8 + 18 {(1)}^{2}$

Step 7.2.1.3

One to any power is one.

$f' (1) = 8 + 18 \cdot 1$

Step 7.2.1.4

Multiply $18$ by $1$ .

$f' (1) = 8 + 18$

Step 7.2.2

Add $8$ and $18$ .

$f' (1) = 26$

Step 7.2.3

The final answer is $26$ .

$26$

Step 7.3

At $x = 1$ the derivative is $26$ . 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: $(- \frac{9}{4}, 0), (0, \infty)$

Decreasing on: $(- \infty, - \frac{9}{4})$

Step 9