Calculus Examples

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

Step 1

Rewrite the right side with rational exponents.

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Step 1.1

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

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

Step 1.2

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

$y = \frac{1}{{(x + x^{\frac{1}{3}})}^{\frac{1}{2}}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{1}{{(x + x^{\frac{1}{3}})}^{\frac{1}{2}}})$

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

Apply basic rules of exponents.

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Step 4.1.1

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

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

Step 4.1.2

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

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Step 4.1.2.1

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

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

Step 4.1.2.2

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

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

Step 4.1.2.3

Move the negative in front of the fraction.

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

Step 4.2

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

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Step 4.2.1

To apply the Chain Rule, set $u$ as $x + x^{\frac{1}{3}}$ .

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

Step 4.2.2

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

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

Step 4.2.3

Replace all occurrences of $u$ with $x + x^{\frac{1}{3}}$ .

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Combine the numerators over the common denominator.

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

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $- 1$ by $2$ .

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

Step 4.6.2

Subtract $2$ from $- 1$ .

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

Step 4.7

Combine fractions.

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Step 4.7.1

Move the negative in front of the fraction.

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

Step 4.7.2

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

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

Step 4.7.3

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

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

Step 4.8

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

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

Step 4.9

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

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

Step 4.10

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

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

Step 4.11

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

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

Step 4.12

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

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

Step 4.13

Combine the numerators over the common denominator.

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

Step 4.14

Simplify the numerator.

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Step 4.14.1

Multiply $- 1$ by $3$ .

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

Step 4.14.2

Subtract $3$ from $1$ .

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

Step 4.15

Move the negative in front of the fraction.

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

Step 4.16

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

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

Step 4.17

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

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

Step 4.18

Simplify.

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Step 4.18.1

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

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

Step 4.18.2

Apply the distributive property.

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

Step 4.18.3

Multiply $- 1$ by $1$ .

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

Step 4.18.4

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

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

Step 4.18.5

Simplify the numerator.

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Step 4.18.5.1

Factor $- 1$ out of $- 1 - \frac{1}{3 x^{\frac{2}{3}}}$ .

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Step 4.18.5.1.1

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

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

Step 4.18.5.1.2

Factor $- 1$ out of $- \frac{1}{3 x^{\frac{2}{3}}}$ .

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

Step 4.18.5.1.3

Factor $- 1$ out of $- 1 (1) - (\frac{1}{3 x^{\frac{2}{3}}})$ .

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

Step 4.18.5.1.4

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

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

Step 4.18.5.2

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

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

Step 4.18.5.3

Combine the numerators over the common denominator.

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

Step 4.18.6

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.18.7

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

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Step 4.18.7.1

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

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

Step 4.18.7.2

Multiply $3$ by $2$ .

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

Step 5

Reform the equation by setting the left side equal to the right side.

$y' = - \frac{3 x^{\frac{2}{3}} + 1}{6 {(x + x^{\frac{1}{3}})}^{\frac{3}{2}} x^{\frac{2}{3}}}$

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = - \frac{3 x^{\frac{2}{3}} + 1}{6 {(x + x^{\frac{1}{3}})}^{\frac{3}{2}} x^{\frac{2}{3}}}$