Calculus Examples

$f (x) = {(4 - x^{2})}^{\frac{2}{3}}$

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 using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{2}{3}}$ and $g (x) = 4 - x^{2}$ .

Tap for more steps...

Step 1.1.1.1

To apply the Chain Rule, set $u$ as $4 - x^{2}$ .

$\frac{d}{d u} [u^{\frac{2}{3}}] \frac{d}{d x} [4 - x^{2}]$

Step 1.1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{2}{3}$ .

$\frac{2}{3} u^{\frac{2}{3} - 1} \frac{d}{d x} [4 - x^{2}]$

Step 1.1.1.3

Replace all occurrences of $u$ with $4 - x^{2}$ .

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

Step 1.1.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 1.1.3

Combine $- 1$ and $\frac{3}{3}$ .

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

Step 1.1.4

Combine the numerators over the common denominator.

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

Step 1.1.5

Simplify the numerator.

Tap for more steps...

Step 1.1.5.1

Multiply $- 1$ by $3$ .

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

Step 1.1.5.2

Subtract $3$ from $2$ .

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

Step 1.1.6

Combine fractions.

Tap for more steps...

Step 1.1.6.1

Move the negative in front of the fraction.

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

Step 1.1.6.2

Combine $\frac{2}{3}$ and ${(4 - x^{2})}^{- \frac{1}{3}}$ .

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

Step 1.1.6.3

Move ${(4 - x^{2})}^{- \frac{1}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.1.7

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

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

Step 1.1.8

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$\frac{2}{3 {(4 - x^{2})}^{\frac{1}{3}}} (0 + \frac{d}{d x} [- x^{2}])$

Step 1.1.9

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

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

Step 1.1.10

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

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

Step 1.1.11

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

$\frac{2}{3 {(4 - x^{2})}^{\frac{1}{3}}} (- (2 x))$

Step 1.1.12

Combine fractions.

Tap for more steps...

Step 1.1.12.1

Multiply $2$ by $- 1$ .

$\frac{2}{3 {(4 - x^{2})}^{\frac{1}{3}}} (- 2 x)$

Step 1.1.12.2

Combine $- 2$ and $\frac{2}{3 {(4 - x^{2})}^{\frac{1}{3}}}$ .

$\frac{- 2 \cdot 2}{3 {(4 - x^{2})}^{\frac{1}{3}}} x$

Step 1.1.12.3

Multiply $- 2$ by $2$ .

$\frac{- 4}{3 {(4 - x^{2})}^{\frac{1}{3}}} x$

Step 1.1.12.4

Combine $\frac{- 4}{3 {(4 - x^{2})}^{\frac{1}{3}}}$ and $x$ .

$\frac{- 4 x}{3 {(4 - x^{2})}^{\frac{1}{3}}}$

Step 1.1.12.5

Move the negative in front of the fraction.

$f' (x) = - \frac{4 x}{3 {(4 - x^{2})}^{\frac{1}{3}}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{4 x}{3 {(4 - x^{2})}^{\frac{1}{3}}}$ .

$- \frac{4 x}{3 {(4 - x^{2})}^{\frac{1}{3}}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{4 x}{3 {(4 - x^{2})}^{\frac{1}{3}}} = 0$ .

Tap for more steps...

Step 2.1

Set the first derivative equal to $0$ .

$- \frac{4 x}{3 {(4 - x^{2})}^{\frac{1}{3}}} = 0$

Step 2.2

Set the numerator equal to zero.

$4 x = 0$

Step 2.3

Divide each term in $4 x = 0$ by $4$ and simplify.

Tap for more steps...

Step 2.3.1

Divide each term in $4 x = 0$ by $4$ .

$\frac{4 x}{4} = \frac{0}{4}$

Step 2.3.2

Simplify the left side.

Tap for more steps...

Step 2.3.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 2.3.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{0}{4}$

Step 2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{4}$

Step 2.3.3

Simplify the right side.

Tap for more steps...

Step 2.3.3.1

Divide $0$ by $4$ .

$x = 0$

Step 3

Find the values where the derivative is undefined.

Tap for more steps...

Step 3.1

Convert expressions with fractional exponents to radicals.

Tap for more steps...

Step 3.1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$- \frac{4 x}{3 \sqrt[3]{{(4 - x^{2})}^{1}}}$

Step 3.1.2

Anything raised to $1$ is the base itself.

$- \frac{4 x}{3 \sqrt[3]{4 - x^{2}}}$

Step 3.2

Set the denominator in $\frac{4 x}{3 \sqrt[3]{4 - x^{2}}}$ equal to $0$ to find where the expression is undefined.

$3 \sqrt[3]{4 - x^{2}} = 0$

Step 3.3

Solve for $x$ .

Tap for more steps...

Step 3.3.1

To remove the radical on the left side of the equation, cube both sides of the equation.

${(3 \sqrt[3]{4 - x^{2}})}^{3} = 0^{3}$

Step 3.3.2

Simplify each side of the equation.

Tap for more steps...

Step 3.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4 - x^{2}}$ as ${(4 - x^{2})}^{\frac{1}{3}}$ .

${(3 {(4 - x^{2})}^{\frac{1}{3}})}^{3} = 0^{3}$

Step 3.3.2.2

Simplify the left side.

Tap for more steps...

Step 3.3.2.2.1

Simplify ${(3 {(4 - x^{2})}^{\frac{1}{3}})}^{3}$ .

Tap for more steps...

Step 3.3.2.2.1.1

Apply the product rule to $3 {(4 - x^{2})}^{\frac{1}{3}}$ .

$3^{3} {({(4 - x^{2})}^{\frac{1}{3}})}^{3} = 0^{3}$

Step 3.3.2.2.1.2

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

$27 {({(4 - x^{2})}^{\frac{1}{3}})}^{3} = 0^{3}$

Step 3.3.2.2.1.3

Multiply the exponents in ${({(4 - x^{2})}^{\frac{1}{3}})}^{3}$ .

Tap for more steps...

Step 3.3.2.2.1.3.1

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

$27 {(4 - x^{2})}^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 3.3.2.2.1.3.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.3.2.2.1.3.2.1

Cancel the common factor.

$27 {(4 - x^{2})}^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 3.3.2.2.1.3.2.2

Rewrite the expression.

$27 {(4 - x^{2})}^{1} = 0^{3}$

Step 3.3.2.2.1.4

Simplify.

$27 (4 - x^{2}) = 0^{3}$

Step 3.3.2.2.1.5

Apply the distributive property.

$27 \cdot 4 + 27 (- x^{2}) = 0^{3}$

Step 3.3.2.2.1.6

Multiply.

Tap for more steps...

Step 3.3.2.2.1.6.1

Multiply $27$ by $4$ .

$108 + 27 (- x^{2}) = 0^{3}$

Step 3.3.2.2.1.6.2

Multiply $- 1$ by $27$ .

$108 - 27 x^{2} = 0^{3}$

Step 3.3.2.3

Simplify the right side.

Tap for more steps...

Step 3.3.2.3.1

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

$108 - 27 x^{2} = 0$

Step 3.3.3

Solve for $x$ .

Tap for more steps...

Step 3.3.3.1

Subtract $108$ from both sides of the equation.

$- 27 x^{2} = - 108$

Step 3.3.3.2

Divide each term in $- 27 x^{2} = - 108$ by $- 27$ and simplify.

Tap for more steps...

Step 3.3.3.2.1

Divide each term in $- 27 x^{2} = - 108$ by $- 27$ .

$\frac{- 27 x^{2}}{- 27} = \frac{- 108}{- 27}$

Step 3.3.3.2.2

Simplify the left side.

Tap for more steps...

Step 3.3.3.2.2.1

Cancel the common factor of $- 27$ .

Tap for more steps...

Step 3.3.3.2.2.1.1

Cancel the common factor.

$\frac{- 27 x^{2}}{- 27} = \frac{- 108}{- 27}$

Step 3.3.3.2.2.1.2

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

$x^{2} = \frac{- 108}{- 27}$

Step 3.3.3.2.3

Simplify the right side.

Tap for more steps...

Step 3.3.3.2.3.1

Divide $- 108$ by $- 27$ .

$x^{2} = 4$

Step 3.3.3.3

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

$x = \pm \sqrt{4}$

Step 3.3.3.4

Simplify $\pm \sqrt{4}$ .

Tap for more steps...

Step 3.3.3.4.1

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

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

Step 3.3.3.4.2

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

$x = \pm 2$

Step 3.3.3.5

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

Tap for more steps...

Step 3.3.3.5.1

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

$x = 2$

Step 3.3.3.5.2

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

$x = - 2$

Step 3.3.3.5.3

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

$x = 2, - 2$

Step 3.4

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = - 2, x = 2$

Step 4

Evaluate ${(4 - x^{2})}^{\frac{2}{3}}$ at each $x$ value where the derivative is $0$ or undefined.

Tap for more steps...

Step 4.1

Evaluate at $x = 0$ .

Tap for more steps...

Step 4.1.1

Substitute $0$ for $x$ .

${(4 - {(0)}^{2})}^{\frac{2}{3}}$

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

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

${(4 - 0)}^{\frac{2}{3}}$

Step 4.1.2.1.2

Multiply $- 1$ by $0$ .

${(4 + 0)}^{\frac{2}{3}}$

Step 4.1.2.2

Add $4$ and $0$ .

$4^{\frac{2}{3}}$

Step 4.2

Evaluate at $x = - 2$ .

Tap for more steps...

Step 4.2.1

Substitute $- 2$ for $x$ .

${(4 - {(- 2)}^{2})}^{\frac{2}{3}}$

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 $2$ .

${(4 - 1 \cdot 4)}^{\frac{2}{3}}$

Step 4.2.2.1.2

Multiply $- 1$ by $4$ .

${(4 - 4)}^{\frac{2}{3}}$

Step 4.2.2.2

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 4.2.2.2.1

Subtract $4$ from $4$ .

$0^{\frac{2}{3}}$

Step 4.2.2.2.2

Simplify the expression.

Tap for more steps...

Step 4.2.2.2.2.1

Rewrite $0$ as $0^{3}$ .

${(0^{3})}^{\frac{2}{3}}$

Step 4.2.2.2.2.2

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

$0^{3 (\frac{2}{3})}$

Step 4.2.2.2.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.2.2.2.3.1

Cancel the common factor.

$0^{3 (\frac{2}{3})}$

Step 4.2.2.2.3.2

Rewrite the expression.

$0^{2}$

Step 4.2.2.2.4

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

$0$

Step 4.3

Evaluate at $x = 2$ .

Tap for more steps...

Step 4.3.1

Substitute $2$ for $x$ .

${(4 - {(2)}^{2})}^{\frac{2}{3}}$

Step 4.3.2

Simplify.

Tap for more steps...

Step 4.3.2.1

Simplify each term.

Tap for more steps...

Step 4.3.2.1.1

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

${(4 - 1 \cdot 4)}^{\frac{2}{3}}$

Step 4.3.2.1.2

Multiply $- 1$ by $4$ .

${(4 - 4)}^{\frac{2}{3}}$

Step 4.3.2.2

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 4.3.2.2.1

Subtract $4$ from $4$ .

$0^{\frac{2}{3}}$

Step 4.3.2.2.2

Simplify the expression.

Tap for more steps...

Step 4.3.2.2.2.1

Rewrite $0$ as $0^{3}$ .

${(0^{3})}^{\frac{2}{3}}$

Step 4.3.2.2.2.2

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

$0^{3 (\frac{2}{3})}$

Step 4.3.2.2.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.3.2.2.3.1

Cancel the common factor.

$0^{3 (\frac{2}{3})}$

Step 4.3.2.2.3.2

Rewrite the expression.

$0^{2}$

Step 4.3.2.2.4

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

$0$

Step 4.4

List all of the points.

$(0, 4^{\frac{2}{3}}), (- 2, 0), (2, 0)$

Step 5