Calculus Examples

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

Step 1

Find the first derivative of the function.

Tap for more steps...

Step 1.1

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

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

Step 1.2

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

Tap for more steps...

Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

Multiply $3$ by $5$ .

$15 x^{2} + \frac{d}{d x} [- 3 x^{5}]$

Step 1.3

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

Tap for more steps...

Step 1.3.1

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

$15 x^{2} - 3 \frac{d}{d x} [x^{5}]$

Step 1.3.2

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

$15 x^{2} - 3 (5 x^{4})$

Step 1.3.3

Multiply $5$ by $- 3$ .

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

Step 1.4

Reorder terms.

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

Step 2

Find the second derivative of the function.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $- 15 x^{4} + 15 x^{2}$ with respect to $x$ is $\frac{d}{d x} [- 15 x^{4}] + \frac{d}{d x} [15 x^{2}]$ .

$f'' (x) = \frac{d}{d x} (- 15 x^{4}) + \frac{d}{d x} (15 x^{2})$

Step 2.2

Evaluate $\frac{d}{d x} [- 15 x^{4}]$ .

Tap for more steps...

Step 2.2.1

Since $- 15$ is constant with respect to $x$ , the derivative of $- 15 x^{4}$ with respect to $x$ is $- 15 \frac{d}{d x} [x^{4}]$ .

$f'' (x) = - 15 \frac{d}{d x} x^{4} + \frac{d}{d x} (15 x^{2})$

Step 2.2.2

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

$f'' (x) = - 15 (4 x^{3}) + \frac{d}{d x} (15 x^{2})$

Step 2.2.3

Multiply $4$ by $- 15$ .

$f'' (x) = - 60 x^{3} + \frac{d}{d x} (15 x^{2})$

Step 2.3

Evaluate $\frac{d}{d x} [15 x^{2}]$ .

Tap for more steps...

Step 2.3.1

Since $15$ is constant with respect to $x$ , the derivative of $15 x^{2}$ with respect to $x$ is $15 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = - 60 x^{3} + 15 \frac{d}{d x} (x^{2})$

Step 2.3.2

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

$f'' (x) = - 60 x^{3} + 15 (2 x)$

Step 2.3.3

Multiply $2$ by $15$ .

$f'' (x) = - 60 x^{3} + 30 x$

Step 3

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

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

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

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

$f' (x) = \frac{d}{d x} (5 x^{3}) + \frac{d}{d x} (- 3 x^{5})$

Step 4.1.2

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

Tap for more steps...

Step 4.1.2.1

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

$f' (x) = 5 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 3 x^{5})$

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 $n = 3$ .

$f' (x) = 5 (3 x^{2}) + \frac{d}{d x} (- 3 x^{5})$

Step 4.1.2.3

Multiply $3$ by $5$ .

$f' (x) = 15 x^{2} + \frac{d}{d x} (- 3 x^{5})$

Step 4.1.3

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

Tap for more steps...

Step 4.1.3.1

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

$f' (x) = 15 x^{2} - 3 \frac{d}{d x} x^{5}$

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 $n = 5$ .

$f' (x) = 15 x^{2} - 3 (5 x^{4})$

Step 4.1.3.3

Multiply $5$ by $- 3$ .

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

Step 4.1.4

Reorder terms.

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

Step 4.2

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

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

Step 5

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

Tap for more steps...

Step 5.1

Set the first derivative equal to $0$ .

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

Step 5.2

Factor the left side of the equation.

Tap for more steps...

Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.2.3

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

Tap for more steps...

Step 5.2.3.1

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

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

Step 5.2.3.2

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

$15 u (- u) + 15 u (1) = 0$

Step 5.2.3.3

Factor $15 u$ out of $15 u (- u) + 15 u (1)$ .

$15 u (- u + 1) = 0$

Step 5.2.4

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

$15 x^{2} (- x^{2} + 1) = 0$

Step 5.3

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

$x^{2} = 0$

$- x^{2} + 1 = 0$

Step 5.4

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

Tap for more steps...

Step 5.4.1

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

$x^{2} = 0$

Step 5.4.2

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

Tap for more steps...

Step 5.4.2.1

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

$x = \pm \sqrt{0}$

Step 5.4.2.2

Simplify $\pm \sqrt{0}$ .

Tap for more steps...

Step 5.4.2.2.1

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

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

Step 5.4.2.2.2

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

$x = \pm 0$

Step 5.4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 5.5

Set $- x^{2} + 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 5.5.1

Set $- x^{2} + 1$ equal to $0$ .

$- x^{2} + 1 = 0$

Step 5.5.2

Solve $- x^{2} + 1 = 0$ for $x$ .

Tap for more steps...

Step 5.5.2.1

Subtract $1$ from both sides of the equation.

$- x^{2} = - 1$

Step 5.5.2.2

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

Tap for more steps...

Step 5.5.2.2.1

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

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

Step 5.5.2.2.2

Simplify the left side.

Tap for more steps...

Step 5.5.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 5.5.2.2.2.2

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

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

Step 5.5.2.2.3

Simplify the right side.

Tap for more steps...

Step 5.5.2.2.3.1

Divide $- 1$ by $- 1$ .

$x^{2} = 1$

Step 5.5.2.3

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

$x = \pm \sqrt{1}$

Step 5.5.2.4

Any root of $1$ is $1$ .

$x = \pm 1$

Step 5.5.2.5

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

Tap for more steps...

Step 5.5.2.5.1

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

$x = 1$

Step 5.5.2.5.2

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

$x = - 1$

Step 5.5.2.5.3

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

$x = 1, - 1$

Step 5.6

The final solution is all the values that make $15 x^{2} (- x^{2} + 1) = 0$ true.

$x = 0, 1, - 1$

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.

$x = 0, 1, - 1$

Step 8

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.

$- 60 {(0)}^{3} + 30 (0)$

Step 9

Evaluate the second derivative.

Tap for more steps...

Step 9.1

Simplify each term.

Tap for more steps...

Step 9.1.1

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

$- 60 \cdot 0 + 30 (0)$

Step 9.1.2

Multiply $- 60$ by $0$ .

$0 + 30 (0)$

Step 9.1.3

Multiply $30$ by $0$ .

$0 + 0$

Step 9.2

Add $0$ and $0$ .

$0$

Step 10

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

Tap for more steps...

Step 10.1

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

$(- \infty, - 1) \cup (- 1, 0) \cup (0, 1) \cup (1, \infty)$

Step 10.2

Substitute any number, such as $- 4$ , from the interval $(- \infty, - 1)$ in the first derivative $- 15 x^{4} + 15 x^{2}$ to check if the result is negative or positive.

Tap for more steps...

Step 10.2.1

Replace the variable $x$ with $- 4$ in the expression.

$f' (- 4) = - 15 {(- 4)}^{4} + 15 {(- 4)}^{2}$

Step 10.2.2

Simplify the result.

Tap for more steps...

Step 10.2.2.1

Simplify each term.

Tap for more steps...

Step 10.2.2.1.1

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

$f' (- 4) = - 15 \cdot 256 + 15 {(- 4)}^{2}$

Step 10.2.2.1.2

Multiply $- 15$ by $256$ .

$f' (- 4) = - 3840 + 15 {(- 4)}^{2}$

Step 10.2.2.1.3

Raise $- 4$ to the power of $2$ .

$f' (- 4) = - 3840 + 15 \cdot 16$

Step 10.2.2.1.4

Multiply $15$ by $16$ .

$f' (- 4) = - 3840 + 240$

Step 10.2.2.2

Add $- 3840$ and $240$ .

$f' (- 4) = - 3600$

Step 10.2.2.3

The final answer is $- 3600$ .

$- 3600$

Step 10.3

Substitute any number, such as $- 0.5$ , from the interval $(- 1, 0)$ in the first derivative $- 15 x^{4} + 15 x^{2}$ to check if the result is negative or positive.

Tap for more steps...

Step 10.3.1

Replace the variable $x$ with $- 0.5$ in the expression.

$f' (- 0.5) = - 15 {(- 0.5)}^{4} + 15 {(- 0.5)}^{2}$

Step 10.3.2

Simplify the result.

Tap for more steps...

Step 10.3.2.1

Simplify each term.

Tap for more steps...

Step 10.3.2.1.1

Raise $- 0.5$ to the power of $4$ .

$f' (- 0.5) = - 15 \cdot 0.0625 + 15 {(- 0.5)}^{2}$

Step 10.3.2.1.2

Multiply $- 15$ by $0.0625$ .

$f' (- 0.5) = - 0.9375 + 15 {(- 0.5)}^{2}$

Step 10.3.2.1.3

Raise $- 0.5$ to the power of $2$ .

$f' (- 0.5) = - 0.9375 + 15 \cdot 0.25$

Step 10.3.2.1.4

Multiply $15$ by $0.25$ .

$f' (- 0.5) = - 0.9375 + 3.75$

Step 10.3.2.2

Add $- 0.9375$ and $3.75$ .

$f' (- 0.5) = 2.8125$

Step 10.3.2.3

The final answer is $2.8125$ .

$2.8125$

Step 10.4

Substitute any number, such as $0.5$ , from the interval $(0, 1)$ in the first derivative $- 15 x^{4} + 15 x^{2}$ to check if the result is negative or positive.

Tap for more steps...

Step 10.4.1

Replace the variable $x$ with $0.5$ in the expression.

$f' (0.5) = - 15 {(0.5)}^{4} + 15 {(0.5)}^{2}$

Step 10.4.2

Simplify the result.

Tap for more steps...

Step 10.4.2.1

Simplify each term.

Tap for more steps...

Step 10.4.2.1.1

Raise $0.5$ to the power of $4$ .

$f' (0.5) = - 15 \cdot 0.0625 + 15 {(0.5)}^{2}$

Step 10.4.2.1.2

Multiply $- 15$ by $0.0625$ .

$f' (0.5) = - 0.9375 + 15 {(0.5)}^{2}$

Step 10.4.2.1.3

Raise $0.5$ to the power of $2$ .

$f' (0.5) = - 0.9375 + 15 \cdot 0.25$

Step 10.4.2.1.4

Multiply $15$ by $0.25$ .

$f' (0.5) = - 0.9375 + 3.75$

Step 10.4.2.2

Add $- 0.9375$ and $3.75$ .

$f' (0.5) = 2.8125$

Step 10.4.2.3

The final answer is $2.8125$ .

$2.8125$

Step 10.5

Substitute any number, such as $4$ , from the interval $(1, \infty)$ in the first derivative $- 15 x^{4} + 15 x^{2}$ to check if the result is negative or positive.

Tap for more steps...

Step 10.5.1

Replace the variable $x$ with $4$ in the expression.

$f' (4) = - 15 {(4)}^{4} + 15 {(4)}^{2}$

Step 10.5.2

Simplify the result.

Tap for more steps...

Step 10.5.2.1

Simplify each term.

Tap for more steps...

Step 10.5.2.1.1

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

$f' (4) = - 15 \cdot 256 + 15 {(4)}^{2}$

Step 10.5.2.1.2

Multiply $- 15$ by $256$ .

$f' (4) = - 3840 + 15 {(4)}^{2}$

Step 10.5.2.1.3

Raise $4$ to the power of $2$ .

$f' (4) = - 3840 + 15 \cdot 16$

Step 10.5.2.1.4

Multiply $15$ by $16$ .

$f' (4) = - 3840 + 240$

Step 10.5.2.2

Add $- 3840$ and $240$ .

$f' (4) = - 3600$

Step 10.5.2.3

The final answer is $- 3600$ .

$- 3600$

Step 10.6

Since the first derivative changed signs from negative to positive around $x = - 1$ , then $x = - 1$ is a local minimum.

$x = - 1$ is a local minimum

Step 10.7

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 10.8

Since the first derivative changed signs from positive to negative around $x = 1$ , then $x = 1$ is a local maximum.

$x = 1$ is a local maximum

Step 10.9

These are the local extrema for $f (x) = 5 x^{3} - 3 x^{5}$ .

$x = - 1$ is a local minimum

$x = 1$ is a local maximum

$x = - 1$ is a local minimum

$x = 1$ is a local maximum

Step 11