Calculus Examples

$f (x) = x^{5} - 5 x^{3} - 20 x - 2$

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 $x^{5} - 5 x^{3} - 20 x - 2$ with respect to $x$ is $\frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 5 x^{3}] + \frac{d}{d x} [- 20 x] + \frac{d}{d x} [- 2]$ .

$\frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 5 x^{3}] + \frac{d}{d x} [- 20 x] + \frac{d}{d x} [- 2]$

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 $n = 5$ .

$5 x^{4} + \frac{d}{d x} [- 5 x^{3}] + \frac{d}{d x} [- 20 x] + \frac{d}{d x} [- 2]$

Step 1.1.2

Evaluate $\frac{d}{d x} [- 5 x^{3}]$ .

Tap for more steps...

Step 1.1.2.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x^{3}$ with respect to $x$ is $- 5 \frac{d}{d x} [x^{3}]$ .

$5 x^{4} - 5 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 20 x] + \frac{d}{d x} [- 2]$

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 $n = 3$ .

$5 x^{4} - 5 (3 x^{2}) + \frac{d}{d x} [- 20 x] + \frac{d}{d x} [- 2]$

Step 1.1.2.3

Multiply $3$ by $- 5$ .

$5 x^{4} - 15 x^{2} + \frac{d}{d x} [- 20 x] + \frac{d}{d x} [- 2]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 20 x]$ .

Tap for more steps...

Step 1.1.3.1

Since $- 20$ is constant with respect to $x$ , the derivative of $- 20 x$ with respect to $x$ is $- 20 \frac{d}{d x} [x]$ .

$5 x^{4} - 15 x^{2} - 20 \frac{d}{d x} [x] + \frac{d}{d x} [- 2]$

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 $n = 1$ .

$5 x^{4} - 15 x^{2} - 20 \cdot 1 + \frac{d}{d x} [- 2]$

Step 1.1.3.3

Multiply $- 20$ by $1$ .

$5 x^{4} - 15 x^{2} - 20 + \frac{d}{d x} [- 2]$

Step 1.1.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 1.1.4.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$5 x^{4} - 15 x^{2} - 20 + 0$

Step 1.1.4.2

Add $5 x^{4} - 15 x^{2} - 20$ and $0$ .

$f' (x) = 5 x^{4} - 15 x^{2} - 20$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $5 x^{4} - 15 x^{2} - 20$ .

$5 x^{4} - 15 x^{2} - 20$

Step 2

Set the first derivative equal to $0$ then solve the equation $5 x^{4} - 15 x^{2} - 20 = 0$ .

Tap for more steps...

Step 2.1

Set the first derivative equal to $0$ .

$5 x^{4} - 15 x^{2} - 20 = 0$

Step 2.2

Substitute $u = x^{2}$ into the equation. This will make the quadratic formula easy to use.

$5 u^{2} - 15 u - 20 = 0$

$u = x^{2}$

Step 2.3

Factor the left side of the equation.

Tap for more steps...

Step 2.3.1

Factor $5$ out of $5 u^{2} - 15 u - 20$ .

Tap for more steps...

Step 2.3.1.1

Factor $5$ out of $5 u^{2}$ .

$5 (u^{2}) - 15 u - 20 = 0$

Step 2.3.1.2

Factor $5$ out of $- 15 u$ .

$5 (u^{2}) + 5 (- 3 u) - 20 = 0$

Step 2.3.1.3

Factor $5$ out of $- 20$ .

$5 u^{2} + 5 (- 3 u) + 5 \cdot - 4 = 0$

Step 2.3.1.4

Factor $5$ out of $5 u^{2} + 5 (- 3 u)$ .

$5 (u^{2} - 3 u) + 5 \cdot - 4 = 0$

Step 2.3.1.5

Factor $5$ out of $5 (u^{2} - 3 u) + 5 \cdot - 4$ .

$5 (u^{2} - 3 u - 4) = 0$

Step 2.3.2

Factor.

Tap for more steps...

Step 2.3.2.1

Factor $u^{2} - 3 u - 4$ using the AC method.

Tap for more steps...

Step 2.3.2.1.1

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 $- 4$ and whose sum is $- 3$ .

$- 4, 1$

Step 2.3.2.1.2

Write the factored form using these integers.

$5 ((u - 4) (u + 1)) = 0$

Step 2.3.2.2

Remove unnecessary parentheses.

$5 (u - 4) (u + 1) = 0$

Step 2.4

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

$u - 4 = 0$

$u + 1 = 0$

Step 2.5

Set $u - 4$ equal to $0$ and solve for $u$ .

Tap for more steps...

Step 2.5.1

Set $u - 4$ equal to $0$ .

$u - 4 = 0$

Step 2.5.2

Add $4$ to both sides of the equation.

$u = 4$

Step 2.6

Set $u + 1$ equal to $0$ and solve for $u$ .

Tap for more steps...

Step 2.6.1

Set $u + 1$ equal to $0$ .

$u + 1 = 0$

Step 2.6.2

Subtract $1$ from both sides of the equation.

$u = - 1$

Step 2.7

The final solution is all the values that make $5 (u - 4) (u + 1) = 0$ true.

$u = 4, - 1$

Step 2.8

Substitute the real value of $u = x^{2}$ back into the solved equation.

$x^{2} = 4$

${(x^{2})}^{1} = - 1$

Step 2.9

Solve the first equation for $x$ .

$x^{2} = 4$

Step 2.10

Solve the equation for $x$ .

Tap for more steps...

Step 2.10.1

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

$x = \pm \sqrt{4}$

Step 2.10.2

Simplify $\pm \sqrt{4}$ .

Tap for more steps...

Step 2.10.2.1

Rewrite $4$ as $2^{2}$ .

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

Step 2.10.2.2

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

$x = \pm 2$

Step 2.10.3

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

Tap for more steps...

Step 2.10.3.1

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

$x = 2$

Step 2.10.3.2

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

$x = - 2$

Step 2.10.3.3

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

$x = 2, - 2$

Step 2.11

Solve the second equation for $x$ .

${(x^{2})}^{1} = - 1$

Step 2.12

Solve the equation for $x$ .

Tap for more steps...

Step 2.12.1

Remove parentheses.

$x^{2} = - 1$

Step 2.12.2

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

$x = \pm \sqrt{- 1}$

Step 2.12.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i$

Step 2.12.4

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

Tap for more steps...

Step 2.12.4.1

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

$x = i$

Step 2.12.4.2

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

$x = - i$

Step 2.12.4.3

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

$x = i, - i$

Step 2.13

The solution to $5 x^{4} - 15 x^{2} - 20 = 0$ is $x = 2, - 2, i, - i$ .

$x = 2, - 2, i, - i$

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 $x^{5} - 5 x^{3} - 20 x - 2$ at each $x$ value where the derivative is $0$ or undefined.

Tap for more steps...

Step 4.1

Evaluate at $x = 2$ .

Tap for more steps...

Step 4.1.1

Substitute $2$ for $x$ .

${(2)}^{5} - 5 {(2)}^{3} - 20 \cdot 2 - 2$

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

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

$32 - 5 {(2)}^{3} - 20 \cdot 2 - 2$

Step 4.1.2.1.2

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

$32 - 5 \cdot 8 - 20 \cdot 2 - 2$

Step 4.1.2.1.3

Multiply $- 5$ by $8$ .

$32 - 40 - 20 \cdot 2 - 2$

Step 4.1.2.1.4

Multiply $- 20$ by $2$ .

$32 - 40 - 40 - 2$

Step 4.1.2.2

Simplify by subtracting numbers.

Tap for more steps...

Step 4.1.2.2.1

Subtract $40$ from $32$ .

$- 8 - 40 - 2$

Step 4.1.2.2.2

Subtract $40$ from $- 8$ .

$- 48 - 2$

Step 4.1.2.2.3

Subtract $2$ from $- 48$ .

$- 50$

Step 4.2

Evaluate at $x = - 2$ .

Tap for more steps...

Step 4.2.1

Substitute $- 2$ for $x$ .

${(- 2)}^{5} - 5 {(- 2)}^{3} - 20 \cdot - 2 - 2$

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

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

$- 32 - 5 {(- 2)}^{3} - 20 \cdot - 2 - 2$

Step 4.2.2.1.2

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

$- 32 - 5 \cdot - 8 - 20 \cdot - 2 - 2$

Step 4.2.2.1.3

Multiply $- 5$ by $- 8$ .

$- 32 + 40 - 20 \cdot - 2 - 2$

Step 4.2.2.1.4

Multiply $- 20$ by $- 2$ .

$- 32 + 40 + 40 - 2$

Step 4.2.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 4.2.2.2.1

Add $- 32$ and $40$ .

$8 + 40 - 2$

Step 4.2.2.2.2

Add $8$ and $40$ .

$48 - 2$

Step 4.2.2.2.3

Subtract $2$ from $48$ .

$46$

Step 4.3

List all of the points.

$(2, - 50), (- 2, 46)$

Step 5