Calculus Examples

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

Step 1

Consider the limit definition of the derivative.

$f' (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$

Step 2

Find the components of the definition.

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Step 2.1

Evaluate the function at $x = x + h$ .

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Step 2.1.1

Replace the variable $x$ with $x + h$ in the expression.

$f (x + h) = {(x + h)}^{\frac{2}{3}}$

Step 2.1.2

The final answer is ${(x + h)}^{\frac{2}{3}}$ .

${(x + h)}^{\frac{2}{3}}$

Step 2.2

Find the components of the definition.

$f (x + h) = {(x + h)}^{\frac{2}{3}}$

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

$f (x + h) = {(x + h)}^{\frac{2}{3}}$

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

Step 3

Plug in the components.

$f' (x) = lim_{h \to 0} \frac{{(x + h)}^{\frac{2}{3}} - (x^{\frac{2}{3}})}{h}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite ${(x + h)}^{\frac{2}{3}}$ as ${({(x + h)}^{\frac{1}{3}})}^{2}$ .

$f' (x) = lim_{h \to 0} \frac{{({(x + h)}^{\frac{1}{3}})}^{2} - x^{\frac{2}{3}}}{h}$

Step 4.2

Rewrite $x^{\frac{2}{3}}$ as ${(x^{\frac{1}{3}})}^{2}$ .

$f' (x) = lim_{h \to 0} \frac{{({(x + h)}^{\frac{1}{3}})}^{2} - {(x^{\frac{1}{3}})}^{2}}{h}$

Step 4.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = {(x + h)}^{\frac{1}{3}}$ and $b = x^{\frac{1}{3}}$ .

$f' (x) = lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + x^{\frac{1}{3}}) ({(x + h)}^{\frac{1}{3}} - x^{\frac{1}{3}})}{h}$

Step 5

Convert fractional exponents to radicals.

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Step 5.1

Rewrite ${(x + h)}^{\frac{1}{3}}$ as $\sqrt[3]{x + h}$ .

$lim_{h \to 0} \frac{(\sqrt[3]{x + h} + x^{\frac{1}{3}}) ({(x + h)}^{\frac{1}{3}} - x^{\frac{1}{3}})}{h}$

Step 5.2

Rewrite $x^{\frac{1}{3}}$ as $\sqrt[3]{x}$ .

$lim_{h \to 0} \frac{(\sqrt[3]{x + h} + \sqrt[3]{x}) ({(x + h)}^{\frac{1}{3}} - x^{\frac{1}{3}})}{h}$

Step 5.3

Rewrite ${(x + h)}^{\frac{1}{3}}$ as $\sqrt[3]{x + h}$ .

$lim_{h \to 0} \frac{(\sqrt[3]{x + h} + \sqrt[3]{x}) (\sqrt[3]{x + h} - x^{\frac{1}{3}})}{h}$

Step 5.4

Rewrite $x^{\frac{1}{3}}$ as $\sqrt[3]{x}$ .

$lim_{h \to 0} \frac{(\sqrt[3]{x + h} + \sqrt[3]{x}) (\sqrt[3]{x + h} - \sqrt[3]{x})}{h}$

Step 6

Apply L'Hospital's rule.

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Step 6.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 6.1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{h \to 0} (\sqrt[3]{x + h} + \sqrt[3]{x}) (\sqrt[3]{x + h} - \sqrt[3]{x})}{lim_{h \to 0} h}$

Step 6.1.2

Evaluate the limit of the numerator.

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Step 6.1.2.1

Split the limit using the Product of Limits Rule on the limit as $h$ approaches $0$ .

$\frac{lim_{h \to 0} \sqrt[3]{x + h} + \sqrt[3]{x} \cdot lim_{h \to 0} \sqrt[3]{x + h} - \sqrt[3]{x}}{lim_{h \to 0} h}$

Step 6.1.2.2

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$\frac{(lim_{h \to 0} \sqrt[3]{x + h} + lim_{h \to 0} \sqrt[3]{x}) \cdot lim_{h \to 0} \sqrt[3]{x + h} - \sqrt[3]{x}}{lim_{h \to 0} h}$

Step 6.1.2.3

Move the limit under the radical sign.

$\frac{(\sqrt[3]{lim_{h \to 0} x + h} + lim_{h \to 0} \sqrt[3]{x}) \cdot lim_{h \to 0} \sqrt[3]{x + h} - \sqrt[3]{x}}{lim_{h \to 0} h}$

Step 6.1.2.4

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$\frac{(\sqrt[3]{lim_{h \to 0} x + lim_{h \to 0} h} + lim_{h \to 0} \sqrt[3]{x}) \cdot lim_{h \to 0} \sqrt[3]{x + h} - \sqrt[3]{x}}{lim_{h \to 0} h}$

Step 6.1.2.5

Evaluate the limit of $x$ which is constant as $h$ approaches $0$ .

$\frac{(\sqrt[3]{x + lim_{h \to 0} h} + lim_{h \to 0} \sqrt[3]{x}) \cdot lim_{h \to 0} \sqrt[3]{x + h} - \sqrt[3]{x}}{lim_{h \to 0} h}$

Step 6.1.2.6

Evaluate the limit of $\sqrt[3]{x}$ which is constant as $h$ approaches $0$ .

$\frac{(\sqrt[3]{x + lim_{h \to 0} h} + \sqrt[3]{x}) \cdot lim_{h \to 0} \sqrt[3]{x + h} - \sqrt[3]{x}}{lim_{h \to 0} h}$

Step 6.1.2.7

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$\frac{(\sqrt[3]{x + lim_{h \to 0} h} + \sqrt[3]{x}) \cdot (lim_{h \to 0} \sqrt[3]{x + h} - lim_{h \to 0} \sqrt[3]{x})}{lim_{h \to 0} h}$

Step 6.1.2.8

Move the limit under the radical sign.

$\frac{(\sqrt[3]{x + lim_{h \to 0} h} + \sqrt[3]{x}) \cdot (\sqrt[3]{lim_{h \to 0} x + h} - lim_{h \to 0} \sqrt[3]{x})}{lim_{h \to 0} h}$

Step 6.1.2.9

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$\frac{(\sqrt[3]{x + lim_{h \to 0} h} + \sqrt[3]{x}) \cdot (\sqrt[3]{lim_{h \to 0} x + lim_{h \to 0} h} - lim_{h \to 0} \sqrt[3]{x})}{lim_{h \to 0} h}$

Step 6.1.2.10

Evaluate the limit of $x$ which is constant as $h$ approaches $0$ .

$\frac{(\sqrt[3]{x + lim_{h \to 0} h} + \sqrt[3]{x}) \cdot (\sqrt[3]{x + lim_{h \to 0} h} - lim_{h \to 0} \sqrt[3]{x})}{lim_{h \to 0} h}$

Step 6.1.2.11

Evaluate the limit of $\sqrt[3]{x}$ which is constant as $h$ approaches $0$ .

$\frac{(\sqrt[3]{x + lim_{h \to 0} h} + \sqrt[3]{x}) \cdot (\sqrt[3]{x + lim_{h \to 0} h} - \sqrt[3]{x})}{lim_{h \to 0} h}$

Step 6.1.2.12

Evaluate the limits by plugging in $0$ for all occurrences of $h$ .

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Step 6.1.2.12.1

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$\frac{(\sqrt[3]{x + 0} + \sqrt[3]{x}) \cdot (\sqrt[3]{x + lim_{h \to 0} h} - \sqrt[3]{x})}{lim_{h \to 0} h}$

Step 6.1.2.12.2

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$\frac{(\sqrt[3]{x + 0} + \sqrt[3]{x}) \cdot (\sqrt[3]{x + 0} - \sqrt[3]{x})}{lim_{h \to 0} h}$

Step 6.1.2.13

Simplify the answer.

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Step 6.1.2.13.1

Add $x$ and $0$ .

$\frac{(\sqrt[3]{x} + \sqrt[3]{x}) \cdot (\sqrt[3]{x + 0} - \sqrt[3]{x})}{lim_{h \to 0} h}$

Step 6.1.2.13.2

Add $\sqrt[3]{x}$ and $\sqrt[3]{x}$ .

$\frac{2 \sqrt[3]{x} \cdot (\sqrt[3]{x + 0} - \sqrt[3]{x})}{lim_{h \to 0} h}$

Step 6.1.2.13.3

Combine the opposite terms in $\sqrt[3]{x + 0} - \sqrt[3]{x}$ .

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Step 6.1.2.13.3.1

Add $x$ and $0$ .

$\frac{2 \sqrt[3]{x} \cdot (\sqrt[3]{x} - \sqrt[3]{x})}{lim_{h \to 0} h}$

Step 6.1.2.13.3.2

Subtract $\sqrt[3]{x}$ from $\sqrt[3]{x}$ .

$\frac{2 \sqrt[3]{x} \cdot 0}{lim_{h \to 0} h}$

Step 6.1.2.13.4

Multiply $2 \sqrt[3]{x} \cdot 0$ .

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Step 6.1.2.13.4.1

Multiply $0$ by $2$ .

$\frac{0 \sqrt[3]{x}}{lim_{h \to 0} h}$

Step 6.1.2.13.4.2

Multiply $0$ by $\sqrt[3]{x}$ .

$\frac{0}{lim_{h \to 0} h}$

Step 6.1.3

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$\frac{0}{0}$

Step 6.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 6.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{h \to 0} \frac{(\sqrt[3]{x + h} + \sqrt[3]{x}) (\sqrt[3]{x + h} - \sqrt[3]{x})}{h} = lim_{h \to 0} \frac{\frac{d}{d h} [(\sqrt[3]{x + h} + \sqrt[3]{x}) (\sqrt[3]{x + h} - \sqrt[3]{x})]}{\frac{d}{d h} [h]}$

Step 6.3

Find the derivative of the numerator and denominator.

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Step 6.3.1

Differentiate the numerator and denominator.

$lim_{h \to 0} \frac{\frac{d}{d h} [(\sqrt[3]{x + h} + \sqrt[3]{x}) (\sqrt[3]{x + h} - \sqrt[3]{x})]}{\frac{d}{d h} [h]}$

Step 6.3.2

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

$lim_{h \to 0} \frac{\frac{d}{d h} [({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) (\sqrt[3]{x + h} - \sqrt[3]{x})]}{\frac{d}{d h} [h]}$

Step 6.3.3

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

$lim_{h \to 0} \frac{\frac{d}{d h} [({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x})]}{\frac{d}{d h} [h]}$

Step 6.3.4

Differentiate using the Product Rule which states that $\frac{d}{d h} [f (h) g (h)]$ is $f (h) \frac{d}{d h} [g (h)] + g (h) \frac{d}{d h} [f (h)]$ where $f (h) = {(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}$ and $g (h) = {(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}$ .

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) \frac{d}{d h} [{(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}] + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) \frac{d}{d h} [{(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}]}{\frac{d}{d h} [h]}$

Step 6.3.5

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

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) (\frac{d}{d h} [{(x + h)}^{\frac{1}{3}}] + \frac{d}{d h} [- \sqrt[3]{x}]) + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) \frac{d}{d h} [{(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}]}{\frac{d}{d h} [h]}$

Step 6.3.6

Differentiate using the chain rule, which states that $\frac{d}{d h} [f (g (h))]$ is $f' (g (h)) g' (h)$ where $f (h) = h^{\frac{1}{3}}$ and $g (h) = x + h$ .

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Step 6.3.6.1

To apply the Chain Rule, set $u_{1}$ as $x + h$ .

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) (\frac{d}{d u_{1}} [{u_{1}}^{\frac{1}{3}}] \frac{d}{d h} [x + h] + \frac{d}{d h} [- \sqrt[3]{x}]) + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) \frac{d}{d h} [{(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}]}{\frac{d}{d h} [h]}$

Step 6.3.6.2

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

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) (\frac{1}{3} {u_{1}}^{\frac{1}{3} - 1} \frac{d}{d h} [x + h] + \frac{d}{d h} [- \sqrt[3]{x}]) + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) \frac{d}{d h} [{(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}]}{\frac{d}{d h} [h]}$

Step 6.3.6.3

Replace all occurrences of $u_{1}$ with $x + h$ .

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) (\frac{1}{3} {(x + h)}^{\frac{1}{3} - 1} \frac{d}{d h} [x + h] + \frac{d}{d h} [- \sqrt[3]{x}]) + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) \frac{d}{d h} [{(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}]}{\frac{d}{d h} [h]}$

Step 6.3.7

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

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) (\frac{1}{3} {(x + h)}^{\frac{1}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d h} [x + h] + \frac{d}{d h} [- \sqrt[3]{x}]) + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) \frac{d}{d h} [{(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}]}{\frac{d}{d h} [h]}$

Step 6.3.8

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

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) (\frac{1}{3} {(x + h)}^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d h} [x + h] + \frac{d}{d h} [- \sqrt[3]{x}]) + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) \frac{d}{d h} [{(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}]}{\frac{d}{d h} [h]}$

Step 6.3.9

Combine the numerators over the common denominator.

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) (\frac{1}{3} {(x + h)}^{\frac{1 - 1 \cdot 3}{3}} \frac{d}{d h} [x + h] + \frac{d}{d h} [- \sqrt[3]{x}]) + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) \frac{d}{d h} [{(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}]}{\frac{d}{d h} [h]}$

Step 6.3.10

Simplify the numerator.

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Step 6.3.10.1

Multiply $- 1$ by $3$ .

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) (\frac{1}{3} {(x + h)}^{\frac{1 - 3}{3}} \frac{d}{d h} [x + h] + \frac{d}{d h} [- \sqrt[3]{x}]) + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) \frac{d}{d h} [{(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}]}{\frac{d}{d h} [h]}$

Step 6.3.10.2

Subtract $3$ from $1$ .

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) (\frac{1}{3} {(x + h)}^{\frac{- 2}{3}} \frac{d}{d h} [x + h] + \frac{d}{d h} [- \sqrt[3]{x}]) + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) \frac{d}{d h} [{(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}]}{\frac{d}{d h} [h]}$

Step 6.3.11

Move the negative in front of the fraction.

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) (\frac{1}{3} {(x + h)}^{- \frac{2}{3}} \frac{d}{d h} [x + h] + \frac{d}{d h} [- \sqrt[3]{x}]) + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) \frac{d}{d h} [{(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}]}{\frac{d}{d h} [h]}$

Step 6.3.12

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

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) (\frac{{(x + h)}^{- \frac{2}{3}}}{3} \frac{d}{d h} [x + h] + \frac{d}{d h} [- \sqrt[3]{x}]) + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) \frac{d}{d h} [{(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}]}{\frac{d}{d h} [h]}$

Step 6.3.13

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

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) (\frac{1}{3 {(x + h)}^{\frac{2}{3}}} \frac{d}{d h} [x + h] + \frac{d}{d h} [- \sqrt[3]{x}]) + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) \frac{d}{d h} [{(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}]}{\frac{d}{d h} [h]}$

Step 6.3.14

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

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) (\frac{1}{3 {(x + h)}^{\frac{2}{3}}} (\frac{d}{d h} [x] + \frac{d}{d h} [h]) + \frac{d}{d h} [- \sqrt[3]{x}]) + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) \frac{d}{d h} [{(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}]}{\frac{d}{d h} [h]}$

Step 6.3.15

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

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) (\frac{1}{3 {(x + h)}^{\frac{2}{3}}} (0 + \frac{d}{d h} [h]) + \frac{d}{d h} [- \sqrt[3]{x}]) + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) \frac{d}{d h} [{(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}]}{\frac{d}{d h} [h]}$

Step 6.3.16

Add $0$ and $\frac{d}{d h} [h]$ .

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) (\frac{1}{3 {(x + h)}^{\frac{2}{3}}} \frac{d}{d h} [h] + \frac{d}{d h} [- \sqrt[3]{x}]) + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) \frac{d}{d h} [{(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}]}{\frac{d}{d h} [h]}$

Step 6.3.17

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

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) (\frac{1}{3 {(x + h)}^{\frac{2}{3}}} \cdot 1 + \frac{d}{d h} [- \sqrt[3]{x}]) + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) \frac{d}{d h} [{(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}]}{\frac{d}{d h} [h]}$

Step 6.3.18

Multiply $\frac{1}{3 {(x + h)}^{\frac{2}{3}}}$ by $1$ .

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) (\frac{1}{3 {(x + h)}^{\frac{2}{3}}} + \frac{d}{d h} [- \sqrt[3]{x}]) + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) \frac{d}{d h} [{(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}]}{\frac{d}{d h} [h]}$

Step 6.3.19

Since $- \sqrt[3]{x}$ is constant with respect to $h$ , the derivative of $- \sqrt[3]{x}$ with respect to $h$ is $0$ .

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) (\frac{1}{3 {(x + h)}^{\frac{2}{3}}} + 0) + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) \frac{d}{d h} [{(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}]}{\frac{d}{d h} [h]}$

Step 6.3.20

Add $\frac{1}{3 {(x + h)}^{\frac{2}{3}}}$ and $0$ .

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) \frac{1}{3 {(x + h)}^{\frac{2}{3}}} + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) \frac{d}{d h} [{(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}]}{\frac{d}{d h} [h]}$

Step 6.3.21

By the Sum Rule, the derivative of ${(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}$ with respect to $h$ is $\frac{d}{d h} [{(x + h)}^{\frac{1}{3}}] + \frac{d}{d h} [\sqrt[3]{x}]$ .

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) \frac{1}{3 {(x + h)}^{\frac{2}{3}}} + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) (\frac{d}{d h} [{(x + h)}^{\frac{1}{3}}] + \frac{d}{d h} [\sqrt[3]{x}])}{\frac{d}{d h} [h]}$

Step 6.3.22

Differentiate using the chain rule, which states that $\frac{d}{d h} [f (g (h))]$ is $f' (g (h)) g' (h)$ where $f (h) = h^{\frac{1}{3}}$ and $g (h) = x + h$ .

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Step 6.3.22.1

To apply the Chain Rule, set $u_{2}$ as $x + h$ .

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) \frac{1}{3 {(x + h)}^{\frac{2}{3}}} + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) (\frac{d}{d u_{2}} [{u_{2}}^{\frac{1}{3}}] \frac{d}{d h} [x + h] + \frac{d}{d h} [\sqrt[3]{x}])}{\frac{d}{d h} [h]}$

Step 6.3.22.2

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

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) \frac{1}{3 {(x + h)}^{\frac{2}{3}}} + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) (\frac{1}{3} {u_{2}}^{\frac{1}{3} - 1} \frac{d}{d h} [x + h] + \frac{d}{d h} [\sqrt[3]{x}])}{\frac{d}{d h} [h]}$

Step 6.3.22.3

Replace all occurrences of $u_{2}$ with $x + h$ .

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) \frac{1}{3 {(x + h)}^{\frac{2}{3}}} + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) (\frac{1}{3} {(x + h)}^{\frac{1}{3} - 1} \frac{d}{d h} [x + h] + \frac{d}{d h} [\sqrt[3]{x}])}{\frac{d}{d h} [h]}$

Step 6.3.23

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

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) \frac{1}{3 {(x + h)}^{\frac{2}{3}}} + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) (\frac{1}{3} {(x + h)}^{\frac{1}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d h} [x + h] + \frac{d}{d h} [\sqrt[3]{x}])}{\frac{d}{d h} [h]}$

Step 6.3.24

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

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) \frac{1}{3 {(x + h)}^{\frac{2}{3}}} + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) (\frac{1}{3} {(x + h)}^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d h} [x + h] + \frac{d}{d h} [\sqrt[3]{x}])}{\frac{d}{d h} [h]}$

Step 6.3.25

Combine the numerators over the common denominator.

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) \frac{1}{3 {(x + h)}^{\frac{2}{3}}} + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) (\frac{1}{3} {(x + h)}^{\frac{1 - 1 \cdot 3}{3}} \frac{d}{d h} [x + h] + \frac{d}{d h} [\sqrt[3]{x}])}{\frac{d}{d h} [h]}$

Step 6.3.26

Simplify the numerator.

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Step 6.3.26.1

Multiply $- 1$ by $3$ .

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) \frac{1}{3 {(x + h)}^{\frac{2}{3}}} + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) (\frac{1}{3} {(x + h)}^{\frac{1 - 3}{3}} \frac{d}{d h} [x + h] + \frac{d}{d h} [\sqrt[3]{x}])}{\frac{d}{d h} [h]}$

Step 6.3.26.2

Subtract $3$ from $1$ .

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) \frac{1}{3 {(x + h)}^{\frac{2}{3}}} + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) (\frac{1}{3} {(x + h)}^{\frac{- 2}{3}} \frac{d}{d h} [x + h] + \frac{d}{d h} [\sqrt[3]{x}])}{\frac{d}{d h} [h]}$

Step 6.3.27

Move the negative in front of the fraction.

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) \frac{1}{3 {(x + h)}^{\frac{2}{3}}} + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) (\frac{1}{3} {(x + h)}^{- \frac{2}{3}} \frac{d}{d h} [x + h] + \frac{d}{d h} [\sqrt[3]{x}])}{\frac{d}{d h} [h]}$

Step 6.3.28

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

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) \frac{1}{3 {(x + h)}^{\frac{2}{3}}} + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) (\frac{{(x + h)}^{- \frac{2}{3}}}{3} \frac{d}{d h} [x + h] + \frac{d}{d h} [\sqrt[3]{x}])}{\frac{d}{d h} [h]}$

Step 6.3.29

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

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) \frac{1}{3 {(x + h)}^{\frac{2}{3}}} + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) (\frac{1}{3 {(x + h)}^{\frac{2}{3}}} \frac{d}{d h} [x + h] + \frac{d}{d h} [\sqrt[3]{x}])}{\frac{d}{d h} [h]}$

Step 6.3.30

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

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) \frac{1}{3 {(x + h)}^{\frac{2}{3}}} + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) (\frac{1}{3 {(x + h)}^{\frac{2}{3}}} (\frac{d}{d h} [x] + \frac{d}{d h} [h]) + \frac{d}{d h} [\sqrt[3]{x}])}{\frac{d}{d h} [h]}$

Step 6.3.31

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

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) \frac{1}{3 {(x + h)}^{\frac{2}{3}}} + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) (\frac{1}{3 {(x + h)}^{\frac{2}{3}}} (0 + \frac{d}{d h} [h]) + \frac{d}{d h} [\sqrt[3]{x}])}{\frac{d}{d h} [h]}$

Step 6.3.32

Add $0$ and $\frac{d}{d h} [h]$ .

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) \frac{1}{3 {(x + h)}^{\frac{2}{3}}} + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) (\frac{1}{3 {(x + h)}^{\frac{2}{3}}} \frac{d}{d h} [h] + \frac{d}{d h} [\sqrt[3]{x}])}{\frac{d}{d h} [h]}$

Step 6.3.33

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

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) \frac{1}{3 {(x + h)}^{\frac{2}{3}}} + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) (\frac{1}{3 {(x + h)}^{\frac{2}{3}}} \cdot 1 + \frac{d}{d h} [\sqrt[3]{x}])}{\frac{d}{d h} [h]}$

Step 6.3.34

Multiply $\frac{1}{3 {(x + h)}^{\frac{2}{3}}}$ by $1$ .

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) \frac{1}{3 {(x + h)}^{\frac{2}{3}}} + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) (\frac{1}{3 {(x + h)}^{\frac{2}{3}}} + \frac{d}{d h} [\sqrt[3]{x}])}{\frac{d}{d h} [h]}$

Step 6.3.35

Since $\sqrt[3]{x}$ is constant with respect to $h$ , the derivative of $\sqrt[3]{x}$ with respect to $h$ is $0$ .

Step 6.3.36

Add $\frac{1}{3 {(x + h)}^{\frac{2}{3}}}$ and $0$ .

$lim_{h \to 0} \frac{({(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}) \frac{1}{3 {(x + h)}^{\frac{2}{3}}} + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) \frac{1}{3 {(x + h)}^{\frac{2}{3}}}}{\frac{d}{d h} [h]}$

Step 6.3.37

Simplify.

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Step 6.3.37.1

Simplify each term.

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Step 6.3.37.1.1

Multiply ${(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}$ by $\frac{1}{3 {(x + h)}^{\frac{2}{3}}}$ .

$lim_{h \to 0} \frac{\frac{{(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}}{3 {(x + h)}^{\frac{2}{3}}} + ({(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}) \frac{1}{3 {(x + h)}^{\frac{2}{3}}}}{\frac{d}{d h} [h]}$

Step 6.3.37.1.2

Multiply ${(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}$ by $\frac{1}{3 {(x + h)}^{\frac{2}{3}}}$ .

$lim_{h \to 0} \frac{\frac{{(x + h)}^{\frac{1}{3}} + \sqrt[3]{x}}{3 {(x + h)}^{\frac{2}{3}}} + \frac{{(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}}{3 {(x + h)}^{\frac{2}{3}}}}{\frac{d}{d h} [h]}$

Step 6.3.37.2

Combine the numerators over the common denominator.

$lim_{h \to 0} \frac{\frac{{(x + h)}^{\frac{1}{3}} + \sqrt[3]{x} + {(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}}{3 {(x + h)}^{\frac{2}{3}}}}{\frac{d}{d h} [h]}$

Step 6.3.37.3

Combine the opposite terms in ${(x + h)}^{\frac{1}{3}} + \sqrt[3]{x} + {(x + h)}^{\frac{1}{3}} - \sqrt[3]{x}$ .

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Step 6.3.37.3.1

Subtract $\sqrt[3]{x}$ from $\sqrt[3]{x}$ .

$lim_{h \to 0} \frac{\frac{{(x + h)}^{\frac{1}{3}} + 0 + {(x + h)}^{\frac{1}{3}}}{3 {(x + h)}^{\frac{2}{3}}}}{\frac{d}{d h} [h]}$

Step 6.3.37.3.2

Add ${(x + h)}^{\frac{1}{3}}$ and $0$ .

$lim_{h \to 0} \frac{\frac{{(x + h)}^{\frac{1}{3}} + {(x + h)}^{\frac{1}{3}}}{3 {(x + h)}^{\frac{2}{3}}}}{\frac{d}{d h} [h]}$

Step 6.3.37.4

Add ${(x + h)}^{\frac{1}{3}}$ and ${(x + h)}^{\frac{1}{3}}$ .

$lim_{h \to 0} \frac{\frac{2 {(x + h)}^{\frac{1}{3}}}{3 {(x + h)}^{\frac{2}{3}}}}{\frac{d}{d h} [h]}$

Step 6.3.37.5

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

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

Step 6.3.37.6

Multiply ${(x + h)}^{\frac{2}{3}}$ by ${(x + h)}^{- \frac{1}{3}}$ by adding the exponents.

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Step 6.3.37.6.1

Move ${(x + h)}^{- \frac{1}{3}}$ .

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

Step 6.3.37.6.2

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

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

Step 6.3.37.6.3

Combine the numerators over the common denominator.

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

Step 6.3.37.6.4

Add $- 1$ and $2$ .

$lim_{h \to 0} \frac{\frac{2}{3 {(x + h)}^{\frac{1}{3}}}}{\frac{d}{d h} [h]}$

Step 6.3.38

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

$lim_{h \to 0} \frac{\frac{2}{3 {(x + h)}^{\frac{1}{3}}}}{1}$

Step 6.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{h \to 0} \frac{2}{3 {(x + h)}^{\frac{1}{3}}} \cdot 1$

Step 6.5

Rewrite ${(x + h)}^{\frac{1}{3}}$ as $\sqrt[3]{x + h}$ .

$lim_{h \to 0} \frac{2}{3 \sqrt[3]{x + h}} \cdot 1$

Step 6.6

Multiply $\frac{2}{3 \sqrt[3]{x + h}}$ by $1$ .

$lim_{h \to 0} \frac{2}{3 \sqrt[3]{x + h}}$

Step 7

Evaluate the limit.

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Step 7.1

Move the term $\frac{2}{3}$ outside of the limit because it is constant with respect to $h$ .

$\frac{2}{3} lim_{h \to 0} \frac{1}{\sqrt[3]{x + h}}$

Step 7.2

Split the limit using the Limits Quotient Rule on the limit as $h$ approaches $0$ .

$\frac{2}{3} \cdot \frac{lim_{h \to 0} 1}{lim_{h \to 0} \sqrt[3]{x + h}}$

Step 7.3

Evaluate the limit of $1$ which is constant as $h$ approaches $0$ .

$\frac{2}{3} \cdot \frac{1}{lim_{h \to 0} \sqrt[3]{x + h}}$

Step 7.4

Move the limit under the radical sign.

$\frac{2}{3} \cdot \frac{1}{\sqrt[3]{lim_{h \to 0} x + h}}$

Step 7.5

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$\frac{2}{3} \cdot \frac{1}{\sqrt[3]{lim_{h \to 0} x + lim_{h \to 0} h}}$

Step 7.6

Evaluate the limit of $x$ which is constant as $h$ approaches $0$ .

$\frac{2}{3} \cdot \frac{1}{\sqrt[3]{x + lim_{h \to 0} h}}$

Step 8

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$\frac{2}{3} \cdot \frac{1}{\sqrt[3]{x + 0}}$

Step 9

Simplify the answer.

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Step 9.1

Add $x$ and $0$ .

$\frac{2}{3} \cdot \frac{1}{\sqrt[3]{x}}$

Step 9.2

Multiply $\frac{1}{\sqrt[3]{x}}$ by $\frac{{\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{2}}$ .

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

Step 9.3

Combine and simplify the denominator.

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Step 9.3.1

Multiply $\frac{1}{\sqrt[3]{x}}$ by $\frac{{\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{2}}$ .

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

Step 9.3.2

Raise $\sqrt[3]{x}$ to the power of $1$ .

$\frac{2}{3} \cdot \frac{{\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{1} {\sqrt[3]{x}}^{2}}$

Step 9.3.3

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

$\frac{2}{3} \cdot \frac{{\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{1 + 2}}$

Step 9.3.4

Add $1$ and $2$ .

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

Step 9.3.5

Rewrite ${\sqrt[3]{x}}^{3}$ as $x$ .

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Step 9.3.5.1

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

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

Step 9.3.5.2

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

$\frac{2}{3} \cdot \frac{{\sqrt[3]{x}}^{2}}{x^{\frac{1}{3} \cdot 3}}$

Step 9.3.5.3

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

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

Step 9.3.5.4

Cancel the common factor of $3$ .

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Step 9.3.5.4.1

Cancel the common factor.

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

Step 9.3.5.4.2

Rewrite the expression.

$\frac{2}{3} \cdot \frac{{\sqrt[3]{x}}^{2}}{x^{1}}$

Step 9.3.5.5

Simplify.

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

Step 9.4

Combine.

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

Step 9.5

Rewrite ${\sqrt[3]{x}}^{2}$ as $\sqrt[3]{x^{2}}$ .

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

Step 10