Calculus Examples

$f (x) = 3 x^{4} - 30 x^{2} + 27$

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 $3 x^{4} - 30 x^{2} + 27$ with respect to $x$ is $\frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [- 30 x^{2}] + \frac{d}{d x} [27]$ .

$f' (x) = \frac{d}{d x} (3 x^{4}) + \frac{d}{d x} (- 30 x^{2}) + \frac{d}{d x} (27)$

Step 1.1.2

Evaluate $\frac{d}{d x} [3 x^{4}]$ .

Tap for more steps...

Step 1.1.2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{4}$ with respect to $x$ is $3 \frac{d}{d x} [x^{4}]$ .

$f' (x) = 3 \frac{d}{d x} (x^{4}) + \frac{d}{d x} (- 30 x^{2}) + \frac{d}{d x} (27)$

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 $n = 4$ .

$f' (x) = 3 (4 x^{3}) + \frac{d}{d x} (- 30 x^{2}) + \frac{d}{d x} (27)$

Step 1.1.2.3

Multiply $4$ by $3$ .

$f' (x) = 12 x^{3} + \frac{d}{d x} (- 30 x^{2}) + \frac{d}{d x} (27)$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 30 x^{2}]$ .

Tap for more steps...

Step 1.1.3.1

Since $- 30$ is constant with respect to $x$ , the derivative of $- 30 x^{2}$ with respect to $x$ is $- 30 \frac{d}{d x} [x^{2}]$ .

$f' (x) = 12 x^{3} - 30 \frac{d}{d x} x^{2} + \frac{d}{d x} (27)$

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 $n = 2$ .

$f' (x) = 12 x^{3} - 30 (2 x) + \frac{d}{d x} (27)$

Step 1.1.3.3

Multiply $2$ by $- 30$ .

$f' (x) = 12 x^{3} - 60 x + \frac{d}{d x} (27)$

Step 1.1.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 1.1.4.1

Since $27$ is constant with respect to $x$ , the derivative of $27$ with respect to $x$ is $0$ .

$f' (x) = 12 x^{3} - 60 x + 0$

Step 1.1.4.2

Add $12 x^{3} - 60 x$ and $0$ .

$f' (x) = 12 x^{3} - 60 x$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $12 x^{3} - 60 x$ .

$12 x^{3} - 60 x$

Step 2

Set the first derivative equal to $0$ then solve the equation $12 x^{3} - 60 x = 0$ .

Tap for more steps...

Step 2.1

Set the first derivative equal to $0$ .

$12 x^{3} - 60 x = 0$

Step 2.2

Factor $12 x$ out of $12 x^{3} - 60 x$ .

Tap for more steps...

Step 2.2.1

Factor $12 x$ out of $12 x^{3}$ .

$12 x (x^{2}) - 60 x = 0$

Step 2.2.2

Factor $12 x$ out of $- 60 x$ .

$12 x (x^{2}) + 12 x (- 5) = 0$

Step 2.2.3

Factor $12 x$ out of $12 x (x^{2}) + 12 x (- 5)$ .

$12 x (x^{2} - 5) = 0$

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 $0$ .

$x = 0$

$x^{2} - 5 = 0$

Step 2.4

Set $x$ equal to $0$ .

$x = 0$

Step 2.5

Set $x^{2} - 5$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.5.1

Set $x^{2} - 5$ equal to $0$ .

$x^{2} - 5 = 0$

Step 2.5.2

Solve $x^{2} - 5 = 0$ for $x$ .

Tap for more steps...

Step 2.5.2.1

Add $5$ to both sides of the equation.

$x^{2} = 5$

Step 2.5.2.2

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

$x = \pm \sqrt{5}$

Step 2.5.2.3

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 2.5.2.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{5}$

Step 2.5.2.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{5}$

Step 2.5.2.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{5}, - \sqrt{5}$

Step 2.6

The final solution is all the values that make $12 x (x^{2} - 5) = 0$ true.

$x = 0, \sqrt{5}, - \sqrt{5}$

Step 3

Find the values where the derivative is undefined.

Tap for more steps...

Step 3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

Evaluate $3 x^{4} - 30 x^{2} + 27$ 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 $x$ .

$3 {(0)}^{4} - 30 {(0)}^{2} + 27$

Step 4.1.2

Simplify.

Tap for more steps...

Step 4.1.2.1

Simplify each term.

Tap for more steps...

Step 4.1.2.1.1

Raising $0$ to any positive power yields $0$ .

$3 \cdot 0 - 30 {(0)}^{2} + 27$

Step 4.1.2.1.2

Multiply $3$ by $0$ .

$0 - 30 {(0)}^{2} + 27$

Step 4.1.2.1.3

Raising $0$ to any positive power yields $0$ .

$0 - 30 \cdot 0 + 27$

Step 4.1.2.1.4

Multiply $- 30$ by $0$ .

$0 + 0 + 27$

Step 4.1.2.2

Simplify by adding numbers.

Tap for more steps...

Step 4.1.2.2.1

Add $0$ and $0$ .

$0 + 27$

Step 4.1.2.2.2

Add $0$ and $27$ .

$27$

Step 4.2

Evaluate at $x = \sqrt{5}$ .

Tap for more steps...

Step 4.2.1

Substitute $\sqrt{5}$ for $x$ .

$3 {(\sqrt{5})}^{4} - 30 {(\sqrt{5})}^{2} + 27$

Step 4.2.2

Simplify.

Tap for more steps...

Step 4.2.2.1

Simplify each term.

Tap for more steps...

Step 4.2.2.1.1

Rewrite ${\sqrt{5}}^{4}$ as $5^{2}$ .

Tap for more steps...

Step 4.2.2.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$3 {(5^{\frac{1}{2}})}^{4} - 30 {(\sqrt{5})}^{2} + 27$

Step 4.2.2.1.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$3 \cdot 5^{\frac{1}{2} \cdot 4} - 30 {(\sqrt{5})}^{2} + 27$

Step 4.2.2.1.1.3

Combine $\frac{1}{2}$ and $4$ .

$3 \cdot 5^{\frac{4}{2}} - 30 {(\sqrt{5})}^{2} + 27$

Step 4.2.2.1.1.4

Cancel the common factor of $4$ and $2$ .

Tap for more steps...

Step 4.2.2.1.1.4.1

Factor $2$ out of $4$ .

$3 \cdot 5^{\frac{2 \cdot 2}{2}} - 30 {(\sqrt{5})}^{2} + 27$

Step 4.2.2.1.1.4.2

Cancel the common factors.

Tap for more steps...

Step 4.2.2.1.1.4.2.1

Factor $2$ out of $2$ .

$3 \cdot 5^{\frac{2 \cdot 2}{2 (1)}} - 30 {(\sqrt{5})}^{2} + 27$

Step 4.2.2.1.1.4.2.2

Cancel the common factor.

$3 \cdot 5^{\frac{2 \cdot 2}{2 \cdot 1}} - 30 {(\sqrt{5})}^{2} + 27$

Step 4.2.2.1.1.4.2.3

Rewrite the expression.

$3 \cdot 5^{\frac{2}{1}} - 30 {(\sqrt{5})}^{2} + 27$

Step 4.2.2.1.1.4.2.4

Divide $2$ by $1$ .

$3 \cdot 5^{2} - 30 {(\sqrt{5})}^{2} + 27$

Step 4.2.2.1.2

Raise $5$ to the power of $2$ .

$3 \cdot 25 - 30 {(\sqrt{5})}^{2} + 27$

Step 4.2.2.1.3

Multiply $3$ by $25$ .

$75 - 30 {(\sqrt{5})}^{2} + 27$

Step 4.2.2.1.4

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Tap for more steps...

Step 4.2.2.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$75 - 30 {(5^{\frac{1}{2}})}^{2} + 27$

Step 4.2.2.1.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$75 - 30 \cdot 5^{\frac{1}{2} \cdot 2} + 27$

Step 4.2.2.1.4.3

Combine $\frac{1}{2}$ and $2$ .

$75 - 30 \cdot 5^{\frac{2}{2}} + 27$

Step 4.2.2.1.4.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.2.1.4.4.1

Cancel the common factor.

$75 - 30 \cdot 5^{\frac{2}{2}} + 27$

Step 4.2.2.1.4.4.2

Rewrite the expression.

$75 - 30 \cdot 5^{1} + 27$

Step 4.2.2.1.4.5

Evaluate the exponent.

$75 - 30 \cdot 5 + 27$

Step 4.2.2.1.5

Multiply $- 30$ by $5$ .

$75 - 150 + 27$

Step 4.2.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 4.2.2.2.1

Subtract $150$ from $75$ .

$- 75 + 27$

Step 4.2.2.2.2

Add $- 75$ and $27$ .

$- 48$

Step 4.3

Evaluate at $x = - \sqrt{5}$ .

Tap for more steps...

Step 4.3.1

Substitute $- \sqrt{5}$ for $x$ .

$3 {(- \sqrt{5})}^{4} - 30 {(- \sqrt{5})}^{2} + 27$

Step 4.3.2

Simplify.

Tap for more steps...

Step 4.3.2.1

Simplify each term.

Tap for more steps...

Step 4.3.2.1.1

Apply the product rule to $- \sqrt{5}$ .

$3 ({(- 1)}^{4} {\sqrt{5}}^{4}) - 30 {(- \sqrt{5})}^{2} + 27$

Step 4.3.2.1.2

Raise $- 1$ to the power of $4$ .

$3 (1 {\sqrt{5}}^{4}) - 30 {(- \sqrt{5})}^{2} + 27$

Step 4.3.2.1.3

Multiply ${\sqrt{5}}^{4}$ by $1$ .

$3 {\sqrt{5}}^{4} - 30 {(- \sqrt{5})}^{2} + 27$

Step 4.3.2.1.4

Rewrite ${\sqrt{5}}^{4}$ as $5^{2}$ .

Tap for more steps...

Step 4.3.2.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$3 {(5^{\frac{1}{2}})}^{4} - 30 {(- \sqrt{5})}^{2} + 27$

Step 4.3.2.1.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$3 \cdot 5^{\frac{1}{2} \cdot 4} - 30 {(- \sqrt{5})}^{2} + 27$

Step 4.3.2.1.4.3

Combine $\frac{1}{2}$ and $4$ .

$3 \cdot 5^{\frac{4}{2}} - 30 {(- \sqrt{5})}^{2} + 27$

Step 4.3.2.1.4.4

Cancel the common factor of $4$ and $2$ .

Tap for more steps...

Step 4.3.2.1.4.4.1

Factor $2$ out of $4$ .

$3 \cdot 5^{\frac{2 \cdot 2}{2}} - 30 {(- \sqrt{5})}^{2} + 27$

Step 4.3.2.1.4.4.2

Cancel the common factors.

Tap for more steps...

Step 4.3.2.1.4.4.2.1

Factor $2$ out of $2$ .

$3 \cdot 5^{\frac{2 \cdot 2}{2 (1)}} - 30 {(- \sqrt{5})}^{2} + 27$

Step 4.3.2.1.4.4.2.2

Cancel the common factor.

$3 \cdot 5^{\frac{2 \cdot 2}{2 \cdot 1}} - 30 {(- \sqrt{5})}^{2} + 27$

Step 4.3.2.1.4.4.2.3

Rewrite the expression.

$3 \cdot 5^{\frac{2}{1}} - 30 {(- \sqrt{5})}^{2} + 27$

Step 4.3.2.1.4.4.2.4

Divide $2$ by $1$ .

$3 \cdot 5^{2} - 30 {(- \sqrt{5})}^{2} + 27$

Step 4.3.2.1.5

Raise $5$ to the power of $2$ .

$3 \cdot 25 - 30 {(- \sqrt{5})}^{2} + 27$

Step 4.3.2.1.6

Multiply $3$ by $25$ .

$75 - 30 {(- \sqrt{5})}^{2} + 27$

Step 4.3.2.1.7

Apply the product rule to $- \sqrt{5}$ .

$75 - 30 ({(- 1)}^{2} {\sqrt{5}}^{2}) + 27$

Step 4.3.2.1.8

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

$75 - 30 (1 {\sqrt{5}}^{2}) + 27$

Step 4.3.2.1.9

Multiply ${\sqrt{5}}^{2}$ by $1$ .

$75 - 30 {\sqrt{5}}^{2} + 27$

Step 4.3.2.1.10

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Tap for more steps...

Step 4.3.2.1.10.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$75 - 30 {(5^{\frac{1}{2}})}^{2} + 27$

Step 4.3.2.1.10.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$75 - 30 \cdot 5^{\frac{1}{2} \cdot 2} + 27$

Step 4.3.2.1.10.3

Combine $\frac{1}{2}$ and $2$ .

$75 - 30 \cdot 5^{\frac{2}{2}} + 27$

Step 4.3.2.1.10.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.3.2.1.10.4.1

Cancel the common factor.

$75 - 30 \cdot 5^{\frac{2}{2}} + 27$

Step 4.3.2.1.10.4.2

Rewrite the expression.

$75 - 30 \cdot 5^{1} + 27$

Step 4.3.2.1.10.5

Evaluate the exponent.

$75 - 30 \cdot 5 + 27$

Step 4.3.2.1.11

Multiply $- 30$ by $5$ .

$75 - 150 + 27$

Step 4.3.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 4.3.2.2.1

Subtract $150$ from $75$ .

$- 75 + 27$

Step 4.3.2.2.2

Add $- 75$ and $27$ .

$- 48$

Step 4.4

List all of the points.

$(0, 27), (\sqrt{5}, - 48), (- \sqrt{5}, - 48)$

Step 5