Calculus Examples

$y = 1 + 40 x^{3} - 3 x^{5}$

Step 1

Simplify $1 + 40 x^{3} - 3 x^{5}$ .

Tap for more steps...

Step 1.1

Move $1$ .

$y = 40 x^{3} - 3 x^{5} + 1$

Step 1.2

Reorder $40 x^{3}$ and $- 3 x^{5}$ .

$y = - 3 x^{5} + 40 x^{3} + 1$

Step 2

Set $y$ as a function of $x$ .

$f (x) = - 3 x^{5} + 40 x^{3} + 1$

Step 3

Find the derivative.

Tap for more steps...

Step 3.1

By the Sum Rule, the derivative of $- 3 x^{5} + 40 x^{3} + 1$ with respect to $x$ is $\frac{d}{d x} [- 3 x^{5}] + \frac{d}{d x} [40 x^{3}] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [- 3 x^{5}] + \frac{d}{d x} [40 x^{3}] + \frac{d}{d x} [1]$

Step 3.2

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

Tap for more steps...

Step 3.2.1

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

$- 3 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [40 x^{3}] + \frac{d}{d x} [1]$

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$- 3 (5 x^{4}) + \frac{d}{d x} [40 x^{3}] + \frac{d}{d x} [1]$

Step 3.2.3

Multiply $5$ by $- 3$ .

$- 15 x^{4} + \frac{d}{d x} [40 x^{3}] + \frac{d}{d x} [1]$

Step 3.3

Evaluate $\frac{d}{d x} [40 x^{3}]$ .

Tap for more steps...

Step 3.3.1

Since $40$ is constant with respect to $x$ , the derivative of $40 x^{3}$ with respect to $x$ is $40 \frac{d}{d x} [x^{3}]$ .

$- 15 x^{4} + 40 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [1]$

Step 3.3.2

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

$- 15 x^{4} + 40 (3 x^{2}) + \frac{d}{d x} [1]$

Step 3.3.3

Multiply $3$ by $40$ .

$- 15 x^{4} + 120 x^{2} + \frac{d}{d x} [1]$

Step 3.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 3.4.1

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$- 15 x^{4} + 120 x^{2} + 0$

Step 3.4.2

Add $- 15 x^{4} + 120 x^{2}$ and $0$ .

$- 15 x^{4} + 120 x^{2}$

Step 4

Set the derivative equal to $0$ then solve the equation $- 15 x^{4} + 120 x^{2} = 0$ .

Tap for more steps...

Step 4.1

Factor the left side of the equation.

Tap for more steps...

Step 4.1.1

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$- 15 {(x^{2})}^{2} + 120 x^{2} = 0$

Step 4.1.2

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$- 15 u^{2} + 120 u = 0$

Step 4.1.3

Factor $15 u$ out of $- 15 u^{2} + 120 u$ .

Tap for more steps...

Step 4.1.3.1

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

$15 u (- u) + 120 u = 0$

Step 4.1.3.2

Factor $15 u$ out of $120 u$ .

$15 u (- u) + 15 u (8) = 0$

Step 4.1.3.3

Factor $15 u$ out of $15 u (- u) + 15 u (8)$ .

$15 u (- u + 8) = 0$

Step 4.1.4

Replace all occurrences of $u$ with $x^{2}$ .

$15 x^{2} (- x^{2} + 8) = 0$

Step 4.2

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

$x^{2} = 0$

$- x^{2} + 8 = 0$

Step 4.3

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

Tap for more steps...

Step 4.3.1

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

$x^{2} = 0$

Step 4.3.2

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

Tap for more steps...

Step 4.3.2.1

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

$x = \pm \sqrt{0}$

Step 4.3.2.2

Simplify $\pm \sqrt{0}$ .

Tap for more steps...

Step 4.3.2.2.1

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

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

Step 4.3.2.2.2

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

$x = \pm 0$

Step 4.3.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 4.4

Set $- x^{2} + 8$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.4.1

Set $- x^{2} + 8$ equal to $0$ .

$- x^{2} + 8 = 0$

Step 4.4.2

Solve $- x^{2} + 8 = 0$ for $x$ .

Tap for more steps...

Step 4.4.2.1

Subtract $8$ from both sides of the equation.

$- x^{2} = - 8$

Step 4.4.2.2

Divide each term in $- x^{2} = - 8$ by $- 1$ and simplify.

Tap for more steps...

Step 4.4.2.2.1

Divide each term in $- x^{2} = - 8$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 8}{- 1}$

Step 4.4.2.2.2

Simplify the left side.

Tap for more steps...

Step 4.4.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 8}{- 1}$

Step 4.4.2.2.2.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 8}{- 1}$

Step 4.4.2.2.3

Simplify the right side.

Tap for more steps...

Step 4.4.2.2.3.1

Divide $- 8$ by $- 1$ .

$x^{2} = 8$

Step 4.4.2.3

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

$x = \pm \sqrt{8}$

Step 4.4.2.4

Simplify $\pm \sqrt{8}$ .

Tap for more steps...

Step 4.4.2.4.1

Rewrite $8$ as $2^{2} \cdot 2$ .

Tap for more steps...

Step 4.4.2.4.1.1

Factor $4$ out of $8$ .

$x = \pm \sqrt{4 (2)}$

Step 4.4.2.4.1.2

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

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

Step 4.4.2.4.2

Pull terms out from under the radical.

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

Step 4.4.2.5

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

Tap for more steps...

Step 4.4.2.5.1

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

$x = 2 \sqrt{2}$

Step 4.4.2.5.2

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

$x = - 2 \sqrt{2}$

Step 4.4.2.5.3

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

$x = 2 \sqrt{2}, - 2 \sqrt{2}$

Step 4.5

The final solution is all the values that make $15 x^{2} (- x^{2} + 8) = 0$ true.

$x = 0, 2 \sqrt{2}, - 2 \sqrt{2}$

Step 5

Solve the original function $f (x) = - 3 x^{5} + 40 x^{3} + 1$ at $x = 0$ .

Tap for more steps...

Step 5.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = - 3 {(0)}^{5} + 40 {(0)}^{3} + 1$

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

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

$f (0) = - 3 \cdot 0 + 40 {(0)}^{3} + 1$

Step 5.2.1.2

Multiply $- 3$ by $0$ .

$f (0) = 0 + 40 {(0)}^{3} + 1$

Step 5.2.1.3

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

$f (0) = 0 + 40 \cdot 0 + 1$

Step 5.2.1.4

Multiply $40$ by $0$ .

$f (0) = 0 + 0 + 1$

Step 5.2.2

Simplify by adding numbers.

Tap for more steps...

Step 5.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 1$

Step 5.2.2.2

Add $0$ and $1$ .

$f (0) = 1$

Step 5.2.3

The final answer is $1$ .

$1$

Step 6

Solve the original function $f (x) = - 3 x^{5} + 40 x^{3} + 1$ at $x = 2 \sqrt{2}$ .

Tap for more steps...

Step 6.1

Replace the variable $x$ with $2 \sqrt{2}$ in the expression.

$f (2 \sqrt{2}) = - 3 {(2 \sqrt{2})}^{5} + 40 {(2 \sqrt{2})}^{3} + 1$

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

Apply the product rule to $2 \sqrt{2}$ .

$f (2 \sqrt{2}) = - 3 (2^{5} {\sqrt{2}}^{5}) + 40 {(2 \sqrt{2})}^{3} + 1$

Step 6.2.1.2

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

$f (2 \sqrt{2}) = - 3 (32 {\sqrt{2}}^{5}) + 40 {(2 \sqrt{2})}^{3} + 1$

Step 6.2.1.3

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

$f (2 \sqrt{2}) = - 3 (32 \sqrt{2^{5}}) + 40 {(2 \sqrt{2})}^{3} + 1$

Step 6.2.1.4

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

$f (2 \sqrt{2}) = - 3 (32 \sqrt{32}) + 40 {(2 \sqrt{2})}^{3} + 1$

Step 6.2.1.5

Rewrite $32$ as $4^{2} \cdot 2$ .

Tap for more steps...

Step 6.2.1.5.1

Factor $16$ out of $32$ .

$f (2 \sqrt{2}) = - 3 (32 \sqrt{16 (2)}) + 40 {(2 \sqrt{2})}^{3} + 1$

Step 6.2.1.5.2

Rewrite $16$ as $4^{2}$ .

$f (2 \sqrt{2}) = - 3 (32 \sqrt{4^{2} \cdot 2}) + 40 {(2 \sqrt{2})}^{3} + 1$

Step 6.2.1.6

Pull terms out from under the radical.

$f (2 \sqrt{2}) = - 3 (32 (4 \sqrt{2})) + 40 {(2 \sqrt{2})}^{3} + 1$

Step 6.2.1.7

Multiply $4$ by $32$ .

$f (2 \sqrt{2}) = - 3 (128 \sqrt{2}) + 40 {(2 \sqrt{2})}^{3} + 1$

Step 6.2.1.8

Multiply $128$ by $- 3$ .

$f (2 \sqrt{2}) = - 384 \sqrt{2} + 40 {(2 \sqrt{2})}^{3} + 1$

Step 6.2.1.9

Apply the product rule to $2 \sqrt{2}$ .

$f (2 \sqrt{2}) = - 384 \sqrt{2} + 40 (2^{3} {\sqrt{2}}^{3}) + 1$

Step 6.2.1.10

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

$f (2 \sqrt{2}) = - 384 \sqrt{2} + 40 (8 {\sqrt{2}}^{3}) + 1$

Step 6.2.1.11

Rewrite ${\sqrt{2}}^{3}$ as $\sqrt{2^{3}}$ .

$f (2 \sqrt{2}) = - 384 \sqrt{2} + 40 (8 \sqrt{2^{3}}) + 1$

Step 6.2.1.12

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

$f (2 \sqrt{2}) = - 384 \sqrt{2} + 40 (8 \sqrt{8}) + 1$

Step 6.2.1.13

Rewrite $8$ as $2^{2} \cdot 2$ .

Tap for more steps...

Step 6.2.1.13.1

Factor $4$ out of $8$ .

$f (2 \sqrt{2}) = - 384 \sqrt{2} + 40 (8 \sqrt{4 (2)}) + 1$

Step 6.2.1.13.2

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

$f (2 \sqrt{2}) = - 384 \sqrt{2} + 40 (8 \sqrt{2^{2} \cdot 2}) + 1$

Step 6.2.1.14

Pull terms out from under the radical.

$f (2 \sqrt{2}) = - 384 \sqrt{2} + 40 (8 (2 \sqrt{2})) + 1$

Step 6.2.1.15

Multiply $2$ by $8$ .

$f (2 \sqrt{2}) = - 384 \sqrt{2} + 40 (16 \sqrt{2}) + 1$

Step 6.2.1.16

Multiply $16$ by $40$ .

$f (2 \sqrt{2}) = - 384 \sqrt{2} + 640 \sqrt{2} + 1$

Step 6.2.2

Add $- 384 \sqrt{2}$ and $640 \sqrt{2}$ .

$f (2 \sqrt{2}) = 256 \sqrt{2} + 1$

Step 6.2.3

The final answer is $256 \sqrt{2} + 1$ .

$256 \sqrt{2} + 1$

Step 7

Solve the original function $f (x) = - 3 x^{5} + 40 x^{3} + 1$ at $x = - 2 \sqrt{2}$ .

Tap for more steps...

Step 7.1

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

$f (- 2 \sqrt{2}) = - 3 {(- 2 \sqrt{2})}^{5} + 40 {(- 2 \sqrt{2})}^{3} + 1$

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

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

$f (- 2 \sqrt{2}) = - 3 ({(- 2)}^{5} {\sqrt{2}}^{5}) + 40 {(- 2 \sqrt{2})}^{3} + 1$

Step 7.2.1.2

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

$f (- 2 \sqrt{2}) = - 3 (- 32 {\sqrt{2}}^{5}) + 40 {(- 2 \sqrt{2})}^{3} + 1$

Step 7.2.1.3

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

$f (- 2 \sqrt{2}) = - 3 (- 32 \sqrt{2^{5}}) + 40 {(- 2 \sqrt{2})}^{3} + 1$

Step 7.2.1.4

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

$f (- 2 \sqrt{2}) = - 3 (- 32 \sqrt{32}) + 40 {(- 2 \sqrt{2})}^{3} + 1$

Step 7.2.1.5

Rewrite $32$ as $4^{2} \cdot 2$ .

Tap for more steps...

Step 7.2.1.5.1

Factor $16$ out of $32$ .

$f (- 2 \sqrt{2}) = - 3 (- 32 \sqrt{16 (2)}) + 40 {(- 2 \sqrt{2})}^{3} + 1$

Step 7.2.1.5.2

Rewrite $16$ as $4^{2}$ .

$f (- 2 \sqrt{2}) = - 3 (- 32 \sqrt{4^{2} \cdot 2}) + 40 {(- 2 \sqrt{2})}^{3} + 1$

Step 7.2.1.6

Pull terms out from under the radical.

$f (- 2 \sqrt{2}) = - 3 (- 32 (4 \sqrt{2})) + 40 {(- 2 \sqrt{2})}^{3} + 1$

Step 7.2.1.7

Multiply $4$ by $- 32$ .

$f (- 2 \sqrt{2}) = - 3 (- 128 \sqrt{2}) + 40 {(- 2 \sqrt{2})}^{3} + 1$

Step 7.2.1.8

Multiply $- 128$ by $- 3$ .

$f (- 2 \sqrt{2}) = 384 \sqrt{2} + 40 {(- 2 \sqrt{2})}^{3} + 1$

Step 7.2.1.9

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

$f (- 2 \sqrt{2}) = 384 \sqrt{2} + 40 ({(- 2)}^{3} {\sqrt{2}}^{3}) + 1$

Step 7.2.1.10

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

$f (- 2 \sqrt{2}) = 384 \sqrt{2} + 40 (- 8 {\sqrt{2}}^{3}) + 1$

Step 7.2.1.11

Rewrite ${\sqrt{2}}^{3}$ as $\sqrt{2^{3}}$ .

$f (- 2 \sqrt{2}) = 384 \sqrt{2} + 40 (- 8 \sqrt{2^{3}}) + 1$

Step 7.2.1.12

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

$f (- 2 \sqrt{2}) = 384 \sqrt{2} + 40 (- 8 \sqrt{8}) + 1$

Step 7.2.1.13

Rewrite $8$ as $2^{2} \cdot 2$ .

Tap for more steps...

Step 7.2.1.13.1

Factor $4$ out of $8$ .

$f (- 2 \sqrt{2}) = 384 \sqrt{2} + 40 (- 8 \sqrt{4 (2)}) + 1$

Step 7.2.1.13.2

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

$f (- 2 \sqrt{2}) = 384 \sqrt{2} + 40 (- 8 \sqrt{2^{2} \cdot 2}) + 1$

Step 7.2.1.14

Pull terms out from under the radical.

$f (- 2 \sqrt{2}) = 384 \sqrt{2} + 40 (- 8 (2 \sqrt{2})) + 1$

Step 7.2.1.15

Multiply $2$ by $- 8$ .

$f (- 2 \sqrt{2}) = 384 \sqrt{2} + 40 (- 16 \sqrt{2}) + 1$

Step 7.2.1.16

Multiply $- 16$ by $40$ .

$f (- 2 \sqrt{2}) = 384 \sqrt{2} - 640 \sqrt{2} + 1$

Step 7.2.2

Subtract $640 \sqrt{2}$ from $384 \sqrt{2}$ .

$f (- 2 \sqrt{2}) = - 256 \sqrt{2} + 1$

Step 7.2.3

The final answer is $- 256 \sqrt{2} + 1$ .

$- 256 \sqrt{2} + 1$

Step 8

The horizontal tangent lines on function $f (x) = - 3 x^{5} + 40 x^{3} + 1$ are $y = 1, y = 256 \sqrt{2} + 1, y = - 256 \sqrt{2} + 1$ .

$y = 1, y = 256 \sqrt{2} + 1, y = - 256 \sqrt{2} + 1$

Step 9