Calculus Examples

$f (x) = x (9 - x^{2})$

Step 1

Find the first derivative of the function.

Tap for more steps...

Step 1.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = 9 - x^{2}$ .

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

Step 1.2

Differentiate.

Tap for more steps...

Step 1.2.1

By the Sum Rule, the derivative of $9 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [- x^{2}]$ .

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

Step 1.2.2

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$x (0 + \frac{d}{d x} [- x^{2}]) + (9 - x^{2}) \frac{d}{d x} [x]$

Step 1.2.3

Add $0$ and $\frac{d}{d x} [- x^{2}]$ .

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

Step 1.2.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

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

Step 1.2.5

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

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

Step 1.2.6

Multiply $2$ by $- 1$ .

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

Step 1.3

Raise $x$ to the power of $1$ .

$- 2 (x^{1} x) + (9 - x^{2}) \frac{d}{d x} [x]$

Step 1.4

Raise $x$ to the power of $1$ .

$- 2 (x^{1} x^{1}) + (9 - x^{2}) \frac{d}{d x} [x]$

Step 1.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 2 x^{1 + 1} + (9 - x^{2}) \frac{d}{d x} [x]$

Step 1.6

Add $1$ and $1$ .

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

Step 1.7

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

$- 2 x^{2} + (9 - x^{2}) \cdot 1$

Step 1.8

Simplify by adding terms.

Tap for more steps...

Step 1.8.1

Multiply $9 - x^{2}$ by $1$ .

$- 2 x^{2} + 9 - x^{2}$

Step 1.8.2

Subtract $x^{2}$ from $- 2 x^{2}$ .

$- 3 x^{2} + 9$

Step 2

Find the second derivative of the function.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $- 3 x^{2} + 9$ with respect to $x$ is $\frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [9]$ .

$f'' (x) = \frac{d}{d x} (- 3 x^{2}) + \frac{d}{d x} (9)$

Step 2.2

Evaluate $\frac{d}{d x} [- 3 x^{2}]$ .

Tap for more steps...

Step 2.2.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{2}$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = - 3 \frac{d}{d x} x^{2} + \frac{d}{d x} (9)$

Step 2.2.2

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

$f'' (x) = - 3 (2 x) + \frac{d}{d x} (9)$

Step 2.2.3

Multiply $2$ by $- 3$ .

$f'' (x) = - 6 x + \frac{d}{d x} (9)$

Step 2.3

Differentiate using the Constant Rule.

Tap for more steps...

Step 2.3.1

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$f'' (x) = - 6 x + 0$

Step 2.3.2

Add $- 6 x$ and $0$ .

$f'' (x) = - 6 x$

Step 3

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

$- 3 x^{2} + 9 = 0$

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 using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = 9 - x^{2}$ .

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

Step 4.1.2

Differentiate.

Tap for more steps...

Step 4.1.2.1

By the Sum Rule, the derivative of $9 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [- x^{2}]$ .

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

Step 4.1.2.2

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$x (0 + \frac{d}{d x} [- x^{2}]) + (9 - x^{2}) \frac{d}{d x} [x]$

Step 4.1.2.3

Add $0$ and $\frac{d}{d x} [- x^{2}]$ .

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

Step 4.1.2.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

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

Step 4.1.2.5

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

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

Step 4.1.2.6

Multiply $2$ by $- 1$ .

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

Step 4.1.3

Raise $x$ to the power of $1$ .

$- 2 (x^{1} x) + (9 - x^{2}) \frac{d}{d x} [x]$

Step 4.1.4

Raise $x$ to the power of $1$ .

$- 2 (x^{1} x^{1}) + (9 - x^{2}) \frac{d}{d x} [x]$

Step 4.1.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 2 x^{1 + 1} + (9 - x^{2}) \frac{d}{d x} [x]$

Step 4.1.6

Add $1$ and $1$ .

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

Step 4.1.7

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

$- 2 x^{2} + (9 - x^{2}) \cdot 1$

Step 4.1.8

Simplify by adding terms.

Tap for more steps...

Step 4.1.8.1

Multiply $9 - x^{2}$ by $1$ .

$- 2 x^{2} + 9 - x^{2}$

Step 4.1.8.2

Subtract $x^{2}$ from $- 2 x^{2}$ .

$f' (x) = - 3 x^{2} + 9$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $- 3 x^{2} + 9$ .

$- 3 x^{2} + 9$

Step 5

Set the first derivative equal to $0$ then solve the equation $- 3 x^{2} + 9 = 0$ .

Tap for more steps...

Step 5.1

Set the first derivative equal to $0$ .

$- 3 x^{2} + 9 = 0$

Step 5.2

Subtract $9$ from both sides of the equation.

$- 3 x^{2} = - 9$

Step 5.3

Divide each term in $- 3 x^{2} = - 9$ by $- 3$ and simplify.

Tap for more steps...

Step 5.3.1

Divide each term in $- 3 x^{2} = - 9$ by $- 3$ .

$\frac{- 3 x^{2}}{- 3} = \frac{- 9}{- 3}$

Step 5.3.2

Simplify the left side.

Tap for more steps...

Step 5.3.2.1

Cancel the common factor of $- 3$ .

Tap for more steps...

Step 5.3.2.1.1

Cancel the common factor.

$\frac{- 3 x^{2}}{- 3} = \frac{- 9}{- 3}$

Step 5.3.2.1.2

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

$x^{2} = \frac{- 9}{- 3}$

Step 5.3.3

Simplify the right side.

Tap for more steps...

Step 5.3.3.1

Divide $- 9$ by $- 3$ .

$x^{2} = 3$

Step 5.4

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

$x = \pm \sqrt{3}$

Step 5.5

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

Tap for more steps...

Step 5.5.1

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

$x = \sqrt{3}$

Step 5.5.2

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

$x = - \sqrt{3}$

Step 5.5.3

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

$x = \sqrt{3}, - \sqrt{3}$

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.

$x = \sqrt{3}, - \sqrt{3}$

Step 8

Evaluate the second derivative at $x = \sqrt{3}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 6 \sqrt{3}$

Step 9

$x = \sqrt{3}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \sqrt{3}$ is a local maximum

Step 10

Find the y-value when $x = \sqrt{3}$ .

Tap for more steps...

Step 10.1

Replace the variable $x$ with $\sqrt{3}$ in the expression.

$f (\sqrt{3}) = (\sqrt{3}) (9 - {(\sqrt{3})}^{2})$

Step 10.2

Simplify the result.

Tap for more steps...

Step 10.2.1

Simplify each term.

Tap for more steps...

Step 10.2.1.1

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

Tap for more steps...

Step 10.2.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$f (\sqrt{3}) = \sqrt{3} (9 - {(3^{\frac{1}{2}})}^{2})$

Step 10.2.1.1.2

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

$f (\sqrt{3}) = \sqrt{3} (9 - 3^{\frac{1}{2} \cdot 2})$

Step 10.2.1.1.3

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

$f (\sqrt{3}) = \sqrt{3} (9 - 3^{\frac{2}{2}})$

Step 10.2.1.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 10.2.1.1.4.1

Cancel the common factor.

$f (\sqrt{3}) = \sqrt{3} (9 - 3^{\frac{2}{2}})$

Step 10.2.1.1.4.2

Rewrite the expression.

$f (\sqrt{3}) = \sqrt{3} (9 - 3)$

Step 10.2.1.1.5

Evaluate the exponent.

$f (\sqrt{3}) = \sqrt{3} (9 - 1 \cdot 3)$

Step 10.2.1.2

Multiply $- 1$ by $3$ .

$f (\sqrt{3}) = \sqrt{3} (9 - 3)$

Step 10.2.2

Simplify the expression.

Tap for more steps...

Step 10.2.2.1

Subtract $3$ from $9$ .

$f (\sqrt{3}) = \sqrt{3} \cdot 6$

Step 10.2.2.2

Move $6$ to the left of $\sqrt{3}$ .

$f (\sqrt{3}) = 6 \sqrt{3}$

Step 10.2.3

The final answer is $6 \sqrt{3}$ .

$y = 6 \sqrt{3}$

Step 11

Evaluate the second derivative at $x = - \sqrt{3}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 6 (- \sqrt{3})$

Step 12

Multiply $- 1$ by $- 6$ .

$6 \sqrt{3}$

Step 13

$x = - \sqrt{3}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - \sqrt{3}$ is a local minimum

Step 14

Find the y-value when $x = - \sqrt{3}$ .

Tap for more steps...

Step 14.1

Replace the variable $x$ with $- \sqrt{3}$ in the expression.

$f (- \sqrt{3}) = (- \sqrt{3}) (9 - {(- \sqrt{3})}^{2})$

Step 14.2

Simplify the result.

Tap for more steps...

Step 14.2.1

Simplify each term.

Tap for more steps...

Step 14.2.1.1

Apply the product rule to $- \sqrt{3}$ .

$f (- \sqrt{3}) = - \sqrt{3} (9 - ({(- 1)}^{2} {\sqrt{3}}^{2}))$

Step 14.2.1.2

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

Tap for more steps...

Step 14.2.1.2.1

Move ${(- 1)}^{2}$ .

$f (- \sqrt{3}) = - \sqrt{3} (9 + {(- 1)}^{2} \cdot (- 1 {\sqrt{3}}^{2}))$

Step 14.2.1.2.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

Tap for more steps...

Step 14.2.1.2.2.1

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

$f (- \sqrt{3}) = - \sqrt{3} (9 + {(- 1)}^{2} \cdot ((- 1) {\sqrt{3}}^{2}))$

Step 14.2.1.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- \sqrt{3}) = - \sqrt{3} (9 + {(- 1)}^{2 + 1} {\sqrt{3}}^{2})$

Step 14.2.1.2.3

Add $2$ and $1$ .

$f (- \sqrt{3}) = - \sqrt{3} (9 + {(- 1)}^{3} {\sqrt{3}}^{2})$

Step 14.2.1.3

Raise $- 1$ to the power of $3$ .

$f (- \sqrt{3}) = - \sqrt{3} (9 - {\sqrt{3}}^{2})$

Step 14.2.1.4

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

Tap for more steps...

Step 14.2.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$f (- \sqrt{3}) = - \sqrt{3} (9 - {(3^{\frac{1}{2}})}^{2})$

Step 14.2.1.4.2

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

$f (- \sqrt{3}) = - \sqrt{3} (9 - 3^{\frac{1}{2} \cdot 2})$

Step 14.2.1.4.3

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

$f (- \sqrt{3}) = - \sqrt{3} (9 - 3^{\frac{2}{2}})$

Step 14.2.1.4.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 14.2.1.4.4.1

Cancel the common factor.

$f (- \sqrt{3}) = - \sqrt{3} (9 - 3^{\frac{2}{2}})$

Step 14.2.1.4.4.2

Rewrite the expression.

$f (- \sqrt{3}) = - \sqrt{3} (9 - 3)$

Step 14.2.1.4.5

Evaluate the exponent.

$f (- \sqrt{3}) = - \sqrt{3} (9 - 1 \cdot 3)$

Step 14.2.1.5

Multiply $- 1$ by $3$ .

$f (- \sqrt{3}) = - \sqrt{3} (9 - 3)$

Step 14.2.2

Simplify the expression.

Tap for more steps...

Step 14.2.2.1

Subtract $3$ from $9$ .

$f (- \sqrt{3}) = - \sqrt{3} \cdot 6$

Step 14.2.2.2

Multiply $6$ by $- 1$ .

$f (- \sqrt{3}) = - 6 \sqrt{3}$

Step 14.2.3

The final answer is $- 6 \sqrt{3}$ .

$y = - 6 \sqrt{3}$

Step 15

These are the local extrema for $f (x) = x (9 - x^{2})$ .

$(\sqrt{3}, 6 \sqrt{3})$ is a local maxima

$(- \sqrt{3}, - 6 \sqrt{3})$ is a local minima

Step 16