Calculus Examples

$f (x) = x^{4} - 4 x^{3} + 10 x^{9}$

Step 1

Find the first derivative of the function.

Tap for more steps...

Step 1.1

Differentiate.

Tap for more steps...

Step 1.1.1

By the Sum Rule, the derivative of $x^{4} - 4 x^{3} + 10 x^{9}$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 4 x^{3}] + \frac{d}{d x} [10 x^{9}]$ .

$\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 4 x^{3}] + \frac{d}{d x} [10 x^{9}]$

Step 1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$4 x^{3} + \frac{d}{d x} [- 4 x^{3}] + \frac{d}{d x} [10 x^{9}]$

Step 1.2

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

Tap for more steps...

Step 1.2.1

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

$4 x^{3} - 4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [10 x^{9}]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$4 x^{3} - 4 (3 x^{2}) + \frac{d}{d x} [10 x^{9}]$

Step 1.2.3

Multiply $3$ by $- 4$ .

$4 x^{3} - 12 x^{2} + \frac{d}{d x} [10 x^{9}]$

Step 1.3

Evaluate $\frac{d}{d x} [10 x^{9}]$ .

Tap for more steps...

Step 1.3.1

Since $10$ is constant with respect to $x$ , the derivative of $10 x^{9}$ with respect to $x$ is $10 \frac{d}{d x} [x^{9}]$ .

$4 x^{3} - 12 x^{2} + 10 \frac{d}{d x} [x^{9}]$

Step 1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 9$ .

$4 x^{3} - 12 x^{2} + 10 (9 x^{8})$

Step 1.3.3

Multiply $9$ by $10$ .

$4 x^{3} - 12 x^{2} + 90 x^{8}$

Step 1.4

Reorder terms.

$90 x^{8} + 4 x^{3} - 12 x^{2}$

Step 2

Find the second derivative of the function.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $90 x^{8} + 4 x^{3} - 12 x^{2}$ with respect to $x$ is $\frac{d}{d x} [90 x^{8}] + \frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 12 x^{2}]$ .

$f'' (x) = \frac{d}{d x} (90 x^{8}) + \frac{d}{d x} (4 x^{3}) + \frac{d}{d x} (- 12 x^{2})$

Step 2.2

Evaluate $\frac{d}{d x} [90 x^{8}]$ .

Tap for more steps...

Step 2.2.1

Since $90$ is constant with respect to $x$ , the derivative of $90 x^{8}$ with respect to $x$ is $90 \frac{d}{d x} [x^{8}]$ .

$f'' (x) = 90 \frac{d}{d x} (x^{8}) + \frac{d}{d x} (4 x^{3}) + \frac{d}{d x} (- 12 x^{2})$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 8$ .

$f'' (x) = 90 (8 x^{7}) + \frac{d}{d x} (4 x^{3}) + \frac{d}{d x} (- 12 x^{2})$

Step 2.2.3

Multiply $8$ by $90$ .

$f'' (x) = 720 x^{7} + \frac{d}{d x} (4 x^{3}) + \frac{d}{d x} (- 12 x^{2})$

Step 2.3

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

Tap for more steps...

Step 2.3.1

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

$f'' (x) = 720 x^{7} + 4 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 12 x^{2})$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$f'' (x) = 720 x^{7} + 4 (3 x^{2}) + \frac{d}{d x} (- 12 x^{2})$

Step 2.3.3

Multiply $3$ by $4$ .

$f'' (x) = 720 x^{7} + 12 x^{2} + \frac{d}{d x} (- 12 x^{2})$

Step 2.4

Evaluate $\frac{d}{d x} [- 12 x^{2}]$ .

Tap for more steps...

Step 2.4.1

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 x^{2}$ with respect to $x$ is $- 12 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 720 x^{7} + 12 x^{2} - 12 \frac{d}{d x} x^{2}$

Step 2.4.2

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

$f'' (x) = 720 x^{7} + 12 x^{2} - 12 (2 x)$

Step 2.4.3

Multiply $2$ by $- 12$ .

$f'' (x) = 720 x^{7} + 12 x^{2} - 24 x$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$90 x^{8} + 4 x^{3} - 12 x^{2} = 0$

Step 4

Find the first derivative.

Tap for more steps...

Step 4.1

Find the first derivative.

Tap for more steps...

Step 4.1.1

Differentiate.

Tap for more steps...

Step 4.1.1.1

By the Sum Rule, the derivative of $x^{4} - 4 x^{3} + 10 x^{9}$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 4 x^{3}] + \frac{d}{d x} [10 x^{9}]$ .

$\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 4 x^{3}] + \frac{d}{d x} [10 x^{9}]$

Step 4.1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$4 x^{3} + \frac{d}{d x} [- 4 x^{3}] + \frac{d}{d x} [10 x^{9}]$

Step 4.1.2

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

Tap for more steps...

Step 4.1.2.1

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

$4 x^{3} - 4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [10 x^{9}]$

Step 4.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$4 x^{3} - 4 (3 x^{2}) + \frac{d}{d x} [10 x^{9}]$

Step 4.1.2.3

Multiply $3$ by $- 4$ .

$4 x^{3} - 12 x^{2} + \frac{d}{d x} [10 x^{9}]$

Step 4.1.3

Evaluate $\frac{d}{d x} [10 x^{9}]$ .

Tap for more steps...

Step 4.1.3.1

Since $10$ is constant with respect to $x$ , the derivative of $10 x^{9}$ with respect to $x$ is $10 \frac{d}{d x} [x^{9}]$ .

$4 x^{3} - 12 x^{2} + 10 \frac{d}{d x} [x^{9}]$

Step 4.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 9$ .

$4 x^{3} - 12 x^{2} + 10 (9 x^{8})$

Step 4.1.3.3

Multiply $9$ by $10$ .

$4 x^{3} - 12 x^{2} + 90 x^{8}$

Step 4.1.4

Reorder terms.

$f' (x) = 90 x^{8} + 4 x^{3} - 12 x^{2}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $90 x^{8} + 4 x^{3} - 12 x^{2}$ .

$90 x^{8} + 4 x^{3} - 12 x^{2}$

Step 5

Set the first derivative equal to $0$ then solve the equation $90 x^{8} + 4 x^{3} - 12 x^{2} = 0$ .

Tap for more steps...

Step 5.1

Set the first derivative equal to $0$ .

$90 x^{8} + 4 x^{3} - 12 x^{2} = 0$

Step 5.2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 0.74155776, 0, 0.68455546$

Step 6

Find the values where the derivative is undefined.

Tap for more steps...

Step 6.1

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 7

Critical points to evaluate.

$x = - 0.74155776, 0, 0.68455546$

Step 8

Evaluate the second derivative at $x = - 0.74155776$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$720 {(- 0.74155776)}^{7} + 12 {(- 0.74155776)}^{2} - 24 \cdot - 0.74155776$

Step 9

Evaluate the second derivative.

Tap for more steps...

Step 9.1

Simplify each term.

Tap for more steps...

Step 9.1.1

Raise $- 0.74155776$ to the power of $7$ .

$720 \cdot - 0.12331472 + 12 {(- 0.74155776)}^{2} - 24 \cdot - 0.74155776$

Step 9.1.2

Multiply $720$ by $- 0.12331472$ .

$- 88.78659948 + 12 {(- 0.74155776)}^{2} - 24 \cdot - 0.74155776$

Step 9.1.3

Raise $- 0.74155776$ to the power of $2$ .

$- 88.78659948 + 12 \cdot 0.54990792 - 24 \cdot - 0.74155776$

Step 9.1.4

Multiply $12$ by $0.54990792$ .

$- 88.78659948 + 6.5988951 - 24 \cdot - 0.74155776$

Step 9.1.5

Multiply $- 24$ by $- 0.74155776$ .

$- 88.78659948 + 6.5988951 + 17.79738646$

Step 9.2

Simplify by adding numbers.

Tap for more steps...

Step 9.2.1

Add $- 88.78659948$ and $6.5988951$ .

$- 82.18770438 + 17.79738646$

Step 9.2.2

Add $- 82.18770438$ and $17.79738646$ .

$- 64.39031791$

Step 10

$x = - 0.74155776$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = - 0.74155776$ is a local maximum

Step 11

Find the y-value when $x = - 0.74155776$ .

Tap for more steps...

Step 11.1

Replace the variable $x$ with $- 0.74155776$ in the expression.

$f (- 0.74155776) = {(- 0.74155776)}^{4} - 4 {(- 0.74155776)}^{3} + 10 {(- 0.74155776)}^{9}$

Step 11.2

Simplify the result.

Tap for more steps...

Step 11.2.1

Simplify each term.

Tap for more steps...

Step 11.2.1.1

Raise $- 0.74155776$ to the power of $4$ .

$f (- 0.74155776) = 0.30239872 - 4 {(- 0.74155776)}^{3} + 10 {(- 0.74155776)}^{9}$

Step 11.2.1.2

Raise $- 0.74155776$ to the power of $3$ .

$f (- 0.74155776) = 0.30239872 - 4 \cdot - 0.40778849 + 10 {(- 0.74155776)}^{9}$

Step 11.2.1.3

Multiply $- 4$ by $- 0.40778849$ .

$f (- 0.74155776) = 0.30239872 + 1.63115397 + 10 {(- 0.74155776)}^{9}$

Step 11.2.1.4

Raise $- 0.74155776$ to the power of $9$ .

$f (- 0.74155776) = 0.30239872 + 1.63115397 + 10 \cdot - 0.06781174$

Step 11.2.1.5

Multiply $10$ by $- 0.06781174$ .

$f (- 0.74155776) = 0.30239872 + 1.63115397 - 0.67811742$

Step 11.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 11.2.2.1

Add $0.30239872$ and $1.63115397$ .

$f (- 0.74155776) = 1.9335527 - 0.67811742$

Step 11.2.2.2

Subtract $0.67811742$ from $1.9335527$ .

$f (- 0.74155776) = 1.25543527$

Step 11.2.3

The final answer is $1.25543527$ .

$y = 1.25543527$

Step 12

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$720 {(0)}^{7} + 12 {(0)}^{2} - 24 \cdot 0$

Step 13

Evaluate the second derivative.

Tap for more steps...

Step 13.1

Simplify each term.

Tap for more steps...

Step 13.1.1

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

$720 \cdot 0 + 12 {(0)}^{2} - 24 \cdot 0$

Step 13.1.2

Multiply $720$ by $0$ .

$0 + 12 {(0)}^{2} - 24 \cdot 0$

Step 13.1.3

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

$0 + 12 \cdot 0 - 24 \cdot 0$

Step 13.1.4

Multiply $12$ by $0$ .

$0 + 0 - 24 \cdot 0$

Step 13.1.5

Multiply $- 24$ by $0$ .

$0 + 0 + 0$

Step 13.2

Simplify by adding numbers.

Tap for more steps...

Step 13.2.1

Add $0$ and $0$ .

$0 + 0$

Step 13.2.2

Add $0$ and $0$ .

$0$

Step 14

Since there is at least one point with $0$ or undefined second derivative, apply the first derivative test.

Tap for more steps...

Step 14.1

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

$(- \infty, - 0.74155776) \cup (- 0.74155776, 0) \cup (0, 0.68455546) \cup (0.68455546, \infty)$

Step 14.2

Substitute any number, such as $- 3$ , from the interval $(- \infty, - 0.74155776)$ in the first derivative $90 x^{8} + 4 x^{3} - 12 x^{2}$ to check if the result is negative or positive.

Tap for more steps...

Step 14.2.1

Replace the variable $x$ with $- 3$ in the expression.

$f' (- 3) = 90 {(- 3)}^{8} + 4 {(- 3)}^{3} - 12 {(- 3)}^{2}$

Step 14.2.2

Simplify the result.

Tap for more steps...

Step 14.2.2.1

Simplify each term.

Tap for more steps...

Step 14.2.2.1.1

Raise $- 3$ to the power of $8$ .

$f' (- 3) = 90 \cdot 6561 + 4 {(- 3)}^{3} - 12 {(- 3)}^{2}$

Step 14.2.2.1.2

Multiply $90$ by $6561$ .

$f' (- 3) = 590490 + 4 {(- 3)}^{3} - 12 {(- 3)}^{2}$

Step 14.2.2.1.3

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

$f' (- 3) = 590490 + 4 \cdot - 27 - 12 {(- 3)}^{2}$

Step 14.2.2.1.4

Multiply $4$ by $- 27$ .

$f' (- 3) = 590490 - 108 - 12 {(- 3)}^{2}$

Step 14.2.2.1.5

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

$f' (- 3) = 590490 - 108 - 12 \cdot 9$

Step 14.2.2.1.6

Multiply $- 12$ by $9$ .

$f' (- 3) = 590490 - 108 - 108$

Step 14.2.2.2

Simplify by subtracting numbers.

Tap for more steps...

Step 14.2.2.2.1

Subtract $108$ from $590490$ .

$f' (- 3) = 590382 - 108$

Step 14.2.2.2.2

Subtract $108$ from $590382$ .

$f' (- 3) = 590274$

Step 14.2.2.3

The final answer is $590274$ .

$590274$

Step 14.3

Substitute any number, such as $- 0.37$ , from the interval $(- 0.74155776, 0)$ in the first derivative $90 x^{8} + 4 x^{3} - 12 x^{2}$ to check if the result is negative or positive.

Tap for more steps...

Step 14.3.1

Replace the variable $x$ with $- 0.37$ in the expression.

$f' (- 0.37) = 90 {(- 0.37)}^{8} + 4 {(- 0.37)}^{3} - 12 {(- 0.37)}^{2}$

Step 14.3.2

Simplify the result.

Tap for more steps...

Step 14.3.2.1

Simplify each term.

Tap for more steps...

Step 14.3.2.1.1

Raise $- 0.37$ to the power of $8$ .

$f' (- 0.37) = 90 \cdot 0.00035124 + 4 {(- 0.37)}^{3} - 12 {(- 0.37)}^{2}$

Step 14.3.2.1.2

Multiply $90$ by $0.00035124$ .

$f' (- 0.37) = 0.03161231 + 4 {(- 0.37)}^{3} - 12 {(- 0.37)}^{2}$

Step 14.3.2.1.3

Raise $- 0.37$ to the power of $3$ .

$f' (- 0.37) = 0.03161231 + 4 \cdot - 0.050653 - 12 {(- 0.37)}^{2}$

Step 14.3.2.1.4

Multiply $4$ by $- 0.050653$ .

$f' (- 0.37) = 0.03161231 - 0.202612 - 12 {(- 0.37)}^{2}$

Step 14.3.2.1.5

Raise $- 0.37$ to the power of $2$ .

$f' (- 0.37) = 0.03161231 - 0.202612 - 12 \cdot 0.1369$

Step 14.3.2.1.6

Multiply $- 12$ by $0.1369$ .

$f' (- 0.37) = 0.03161231 - 0.202612 - 1.6428$

Step 14.3.2.2

Simplify by subtracting numbers.

Tap for more steps...

Step 14.3.2.2.1

Subtract $0.202612$ from $0.03161231$ .

$f' (- 0.37) = - 0.17099968 - 1.6428$

Step 14.3.2.2.2

Subtract $1.6428$ from $- 0.17099968$ .

$f' (- 0.37) = - 1.81379968$

Step 14.3.2.3

The final answer is $- 1.81379968$ .

$- 1.81379968$

Step 14.4

Substitute any number, such as $0.34$ , from the interval $(0, 0.68455546)$ in the first derivative $90 x^{8} + 4 x^{3} - 12 x^{2}$ to check if the result is negative or positive.

Tap for more steps...

Step 14.4.1

Replace the variable $x$ with $0.34$ in the expression.

$f' (0.34) = 90 {(0.34)}^{8} + 4 {(0.34)}^{3} - 12 {(0.34)}^{2}$

Step 14.4.2

Simplify the result.

Tap for more steps...

Step 14.4.2.1

Simplify each term.

Tap for more steps...

Step 14.4.2.1.1

Raise $0.34$ to the power of $8$ .

$f' (0.34) = 90 \cdot 0.00017857 + 4 {(0.34)}^{3} - 12 {(0.34)}^{2}$

Step 14.4.2.1.2

Multiply $90$ by $0.00017857$ .

$f' (0.34) = 0.01607214 + 4 {(0.34)}^{3} - 12 {(0.34)}^{2}$

Step 14.4.2.1.3

Raise $0.34$ to the power of $3$ .

$f' (0.34) = 0.01607214 + 4 \cdot 0.039304 - 12 {(0.34)}^{2}$

Step 14.4.2.1.4

Multiply $4$ by $0.039304$ .

$f' (0.34) = 0.01607214 + 0.157216 - 12 {(0.34)}^{2}$

Step 14.4.2.1.5

Raise $0.34$ to the power of $2$ .

$f' (0.34) = 0.01607214 + 0.157216 - 12 \cdot 0.1156$

Step 14.4.2.1.6

Multiply $- 12$ by $0.1156$ .

$f' (0.34) = 0.01607214 + 0.157216 - 1.3872$

Step 14.4.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 14.4.2.2.1

Add $0.01607214$ and $0.157216$ .

$f' (0.34) = 0.17328814 - 1.3872$

Step 14.4.2.2.2

Subtract $1.3872$ from $0.17328814$ .

$f' (0.34) = - 1.21391185$

Step 14.4.2.3

The final answer is $- 1.21391185$ .

$- 1.21391185$

Step 14.5

Substitute any number, such as $3$ , from the interval $(0.68455546, \infty)$ in the first derivative $90 x^{8} + 4 x^{3} - 12 x^{2}$ to check if the result is negative or positive.

Tap for more steps...

Step 14.5.1

Replace the variable $x$ with $3$ in the expression.

$f' (3) = 90 {(3)}^{8} + 4 {(3)}^{3} - 12 {(3)}^{2}$

Step 14.5.2

Simplify the result.

Tap for more steps...

Step 14.5.2.1

Simplify each term.

Tap for more steps...

Step 14.5.2.1.1

Raise $3$ to the power of $8$ .

$f' (3) = 90 \cdot 6561 + 4 {(3)}^{3} - 12 {(3)}^{2}$

Step 14.5.2.1.2

Multiply $90$ by $6561$ .

$f' (3) = 590490 + 4 {(3)}^{3} - 12 {(3)}^{2}$

Step 14.5.2.1.3

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

$f' (3) = 590490 + 4 \cdot 27 - 12 {(3)}^{2}$

Step 14.5.2.1.4

Multiply $4$ by $27$ .

$f' (3) = 590490 + 108 - 12 {(3)}^{2}$

Step 14.5.2.1.5

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

$f' (3) = 590490 + 108 - 12 \cdot 9$

Step 14.5.2.1.6

Multiply $- 12$ by $9$ .

$f' (3) = 590490 + 108 - 108$

Step 14.5.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 14.5.2.2.1

Add $590490$ and $108$ .

$f' (3) = 590598 - 108$

Step 14.5.2.2.2

Subtract $108$ from $590598$ .

$f' (3) = 590490$

Step 14.5.2.3

The final answer is $590490$ .

$590490$

Step 14.6

Since the first derivative changed signs from positive to negative around $x = - 0.74155776$ , then $x = - 0.74155776$ is a local maximum.

$x = - 0.74155776$ is a local maximum

Step 14.7

Since the first derivative did not change signs around $x = 0$ , this is not a local maximum or minimum.

Not a local maximum or minimum

Step 14.8

Since the first derivative changed signs from negative to positive around $x = 0.68455546$ , then $x = 0.68455546$ is a local minimum.

$x = 0.68455546$ is a local minimum

Step 14.9

These are the local extrema for $f (x) = x^{4} - 4 x^{3} + 10 x^{9}$ .

$x = - 0.74155776$ is a local maximum

$x = 0.68455546$ is a local minimum

$x = - 0.74155776$ is a local maximum

$x = 0.68455546$ is a local minimum

Step 15