Calculus Examples

$y = \frac{1}{\sqrt[3]{x + \sqrt{x}}}$

Step 1

Apply basic rules of exponents.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

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

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

Step 1.4.3

Move the negative in front of the fraction.

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- \frac{1}{3}}$ and $g (x) = x + x^{\frac{1}{2}}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $x + x^{\frac{1}{2}}$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $x + x^{\frac{1}{2}}$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $3$ .

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

Step 6.2

Subtract $3$ from $- 1$ .

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

Step 7

Combine fractions.

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Step 7.1

Move the negative in front of the fraction.

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

Step 7.2

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

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

Step 7.3

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

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

Step 8

By the Sum Rule, the derivative of $x + x^{\frac{1}{2}}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [x^{\frac{1}{2}}]$ .

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

Step 9

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

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

Step 10

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

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

Step 11

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 12

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

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

Step 13

Combine the numerators over the common denominator.

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

Step 14

Simplify the numerator.

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Step 14.1

Multiply $- 1$ by $2$ .

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

Step 14.2

Subtract $2$ from $1$ .

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

Step 15

Move the negative in front of the fraction.

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

Step 16

Combine $\frac{1}{2}$ and $x^{- \frac{1}{2}}$ .

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

Step 17

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

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

Step 18

Simplify.

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Step 18.1

Reorder the factors of $- \frac{1}{3 {(x + x^{\frac{1}{2}})}^{\frac{4}{3}}} (1 + \frac{1}{2 x^{\frac{1}{2}}})$ .

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

Step 18.2

Apply the distributive property.

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

Step 18.3

Multiply $- 1$ by $1$ .

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

Step 18.4

Multiply $- 1 - \frac{1}{2 x^{\frac{1}{2}}}$ by $\frac{1}{3 {(x + x^{\frac{1}{2}})}^{\frac{4}{3}}}$ .

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

Step 18.5

Simplify the numerator.

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Step 18.5.1

Factor $- 1$ out of $- 1 - \frac{1}{2 x^{\frac{1}{2}}}$ .

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Step 18.5.1.1

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 18.5.1.2

Factor $- 1$ out of $- \frac{1}{2 x^{\frac{1}{2}}}$ .

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

Step 18.5.1.3

Factor $- 1$ out of $- 1 (1) - (\frac{1}{2 x^{\frac{1}{2}}})$ .

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

Step 18.5.1.4

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

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

Step 18.5.2

Write $1$ as a fraction with a common denominator.

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

Step 18.5.3

Combine the numerators over the common denominator.

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

Step 18.6

Multiply the numerator by the reciprocal of the denominator.

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

Step 18.7

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

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Step 18.7.1

Multiply $\frac{1}{3 {(x + x^{\frac{1}{2}})}^{\frac{4}{3}}}$ by $\frac{2 x^{\frac{1}{2}} + 1}{2 x^{\frac{1}{2}}}$ .

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

Step 18.7.2

Multiply $2$ by $3$ .

$- \frac{2 x^{\frac{1}{2}} + 1}{6 {(x + x^{\frac{1}{2}})}^{\frac{4}{3}} x^{\frac{1}{2}}}$