Calculus Examples

$f (x) = {(4 x - 5)}^{\frac{2}{3}}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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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 $g (x) = 4 x - 5$ .

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Step 1.1.1.1

To apply the Chain Rule, set $u$ as $4 x - 5$ .

$\frac{d}{d u} [u^{\frac{2}{3}}] \frac{d}{d x} [4 x - 5]$

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 $n = \frac{2}{3}$ .

$\frac{2}{3} u^{\frac{2}{3} - 1} \frac{d}{d x} [4 x - 5]$

Step 1.1.1.3

Replace all occurrences of $u$ with $4 x - 5$ .

$\frac{2}{3} {(4 x - 5)}^{\frac{2}{3} - 1} \frac{d}{d x} [4 x - 5]$

Step 1.1.2

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

$\frac{2}{3} {(4 x - 5)}^{\frac{2}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [4 x - 5]$

Step 1.1.3

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

$\frac{2}{3} {(4 x - 5)}^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} [4 x - 5]$

Step 1.1.4

Combine the numerators over the common denominator.

$\frac{2}{3} {(4 x - 5)}^{\frac{2 - 1 \cdot 3}{3}} \frac{d}{d x} [4 x - 5]$

Step 1.1.5

Simplify the numerator.

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Step 1.1.5.1

Multiply $- 1$ by $3$ .

$\frac{2}{3} {(4 x - 5)}^{\frac{2 - 3}{3}} \frac{d}{d x} [4 x - 5]$

Step 1.1.5.2

Subtract $3$ from $2$ .

$\frac{2}{3} {(4 x - 5)}^{\frac{- 1}{3}} \frac{d}{d x} [4 x - 5]$

Step 1.1.6

Combine fractions.

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Step 1.1.6.1

Move the negative in front of the fraction.

$\frac{2}{3} {(4 x - 5)}^{- \frac{1}{3}} \frac{d}{d x} [4 x - 5]$

Step 1.1.6.2

Combine $\frac{2}{3}$ and ${(4 x - 5)}^{- \frac{1}{3}}$ .

$\frac{2 {(4 x - 5)}^{- \frac{1}{3}}}{3} \frac{d}{d x} [4 x - 5]$

Step 1.1.6.3

Move ${(4 x - 5)}^{- \frac{1}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{2}{3 {(4 x - 5)}^{\frac{1}{3}}} \frac{d}{d x} [4 x - 5]$

Step 1.1.7

By the Sum Rule, the derivative of $4 x - 5$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 5]$ .

$\frac{2}{3 {(4 x - 5)}^{\frac{1}{3}}} (\frac{d}{d x} [4 x] + \frac{d}{d x} [- 5])$

Step 1.1.8

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

$\frac{2}{3 {(4 x - 5)}^{\frac{1}{3}}} (4 \frac{d}{d x} [x] + \frac{d}{d x} [- 5])$

Step 1.1.9

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

$\frac{2}{3 {(4 x - 5)}^{\frac{1}{3}}} (4 \cdot 1 + \frac{d}{d x} [- 5])$

Step 1.1.10

Multiply $4$ by $1$ .

$\frac{2}{3 {(4 x - 5)}^{\frac{1}{3}}} (4 + \frac{d}{d x} [- 5])$

Step 1.1.11

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$\frac{2}{3 {(4 x - 5)}^{\frac{1}{3}}} (4 + 0)$

Step 1.1.12

Combine fractions.

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Step 1.1.12.1

Add $4$ and $0$ .

$\frac{2}{3 {(4 x - 5)}^{\frac{1}{3}}} \cdot 4$

Step 1.1.12.2

Combine $\frac{2}{3 {(4 x - 5)}^{\frac{1}{3}}}$ and $4$ .

$\frac{2 \cdot 4}{3 {(4 x - 5)}^{\frac{1}{3}}}$

Step 1.1.12.3

Multiply $2$ by $4$ .

$f' (x) = \frac{8}{3 {(4 x - 5)}^{\frac{1}{3}}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{8}{3 {(4 x - 5)}^{\frac{1}{3}}}$ .

$\frac{8}{3 {(4 x - 5)}^{\frac{1}{3}}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{8}{3 {(4 x - 5)}^{\frac{1}{3}}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{8}{3 {(4 x - 5)}^{\frac{1}{3}}} = 0$

Step 2.2

Set the numerator equal to zero.

$8 = 0$

Step 2.3

Since $8 \neq 0$ , there are no solutions.

No solution

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Convert expressions with fractional exponents to radicals.

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Step 3.1.1

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

$\frac{8}{3 \sqrt[3]{{(4 x - 5)}^{1}}}$

Step 3.1.2

Anything raised to $1$ is the base itself.

$\frac{8}{3 \sqrt[3]{4 x - 5}}$

Step 3.2

Set the denominator in $\frac{8}{3 \sqrt[3]{4 x - 5}}$ equal to $0$ to find where the expression is undefined.

$3 \sqrt[3]{4 x - 5} = 0$

Step 3.3

Solve for $x$ .

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Step 3.3.1

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

${(3 \sqrt[3]{4 x - 5})}^{3} = 0^{3}$

Step 3.3.2

Simplify each side of the equation.

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Step 3.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4 x - 5}$ as ${(4 x - 5)}^{\frac{1}{3}}$ .

${(3 {(4 x - 5)}^{\frac{1}{3}})}^{3} = 0^{3}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Simplify ${(3 {(4 x - 5)}^{\frac{1}{3}})}^{3}$ .

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Step 3.3.2.2.1.1

Apply the product rule to $3 {(4 x - 5)}^{\frac{1}{3}}$ .

$3^{3} {({(4 x - 5)}^{\frac{1}{3}})}^{3} = 0^{3}$

Step 3.3.2.2.1.2

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

$27 {({(4 x - 5)}^{\frac{1}{3}})}^{3} = 0^{3}$

Step 3.3.2.2.1.3

Multiply the exponents in ${({(4 x - 5)}^{\frac{1}{3}})}^{3}$ .

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Step 3.3.2.2.1.3.1

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

$27 {(4 x - 5)}^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 3.3.2.2.1.3.2

Cancel the common factor of $3$ .

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Step 3.3.2.2.1.3.2.1

Cancel the common factor.

$27 {(4 x - 5)}^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 3.3.2.2.1.3.2.2

Rewrite the expression.

$27 {(4 x - 5)}^{1} = 0^{3}$

Step 3.3.2.2.1.4

Simplify.

$27 (4 x - 5) = 0^{3}$

Step 3.3.2.2.1.5

Apply the distributive property.

$27 (4 x) + 27 \cdot - 5 = 0^{3}$

Step 3.3.2.2.1.6

Multiply.

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Step 3.3.2.2.1.6.1

Multiply $4$ by $27$ .

$108 x + 27 \cdot - 5 = 0^{3}$

Step 3.3.2.2.1.6.2

Multiply $27$ by $- 5$ .

$108 x - 135 = 0^{3}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

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

$108 x - 135 = 0$

Step 3.3.3

Solve for $x$ .

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Step 3.3.3.1

Add $135$ to both sides of the equation.

$108 x = 135$

Step 3.3.3.2

Divide each term in $108 x = 135$ by $108$ and simplify.

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Step 3.3.3.2.1

Divide each term in $108 x = 135$ by $108$ .

$\frac{108 x}{108} = \frac{135}{108}$

Step 3.3.3.2.2

Simplify the left side.

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Step 3.3.3.2.2.1

Cancel the common factor of $108$ .

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Step 3.3.3.2.2.1.1

Cancel the common factor.

$\frac{108 x}{108} = \frac{135}{108}$

Step 3.3.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{135}{108}$

Step 3.3.3.2.3

Simplify the right side.

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Step 3.3.3.2.3.1

Cancel the common factor of $135$ and $108$ .

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Step 3.3.3.2.3.1.1

Factor $27$ out of $135$ .

$x = \frac{27 (5)}{108}$

Step 3.3.3.2.3.1.2

Cancel the common factors.

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Step 3.3.3.2.3.1.2.1

Factor $27$ out of $108$ .

$x = \frac{27 \cdot 5}{27 \cdot 4}$

Step 3.3.3.2.3.1.2.2

Cancel the common factor.

$x = \frac{27 \cdot 5}{27 \cdot 4}$

Step 3.3.3.2.3.1.2.3

Rewrite the expression.

$x = \frac{5}{4}$

Step 4

Evaluate ${(4 x - 5)}^{\frac{2}{3}}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = \frac{5}{4}$ .

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Step 4.1.1

Substitute $\frac{5}{4}$ for $x$ .

${(4 (\frac{5}{4}) - 5)}^{\frac{2}{3}}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Cancel the common factor of $4$ .

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Step 4.1.2.1.1

Cancel the common factor.

${(4 (\frac{5}{4}) - 5)}^{\frac{2}{3}}$

Step 4.1.2.1.2

Rewrite the expression.

${(5 - 5)}^{\frac{2}{3}}$

Step 4.1.2.2

Simplify the expression.

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Step 4.1.2.2.1

Subtract $5$ from $5$ .

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

Step 4.1.2.2.2

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

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

Step 4.1.2.2.3

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

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

Step 4.1.2.3

Cancel the common factor of $3$ .

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Step 4.1.2.3.1

Cancel the common factor.

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

Step 4.1.2.3.2

Rewrite the expression.

$0^{2}$

Step 4.1.2.4

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

$0$

Step 4.2

List all of the points.

$(\frac{5}{4}, 0)$

Step 5