Calculus Examples

$f (x) = \frac{x}{\sqrt{x - y}}$

Step 1

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

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

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x$ and $g (x) = {(x - y)}^{\frac{1}{2}}$ .

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Cancel the common factor of $2$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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

Step 4

Simplify.

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

Step 5

Differentiate using the Power Rule.

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Step 5.1

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

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

Step 5.2

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

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

Step 6

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 $g (x) = x - y$ .

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Step 6.1

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

$\frac{{(x - y)}^{\frac{1}{2}} - x (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [x - y])}{x - y}$

Step 6.2

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

$\frac{{(x - y)}^{\frac{1}{2}} - x (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [x - y])}{x - y}$

Step 6.3

Replace all occurrences of $u$ with $x - y$ .

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

Step 7

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

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

Step 8

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

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

Step 9

Combine the numerators over the common denominator.

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

Step 10

Simplify the numerator.

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Step 10.1

Multiply $- 1$ by $2$ .

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

Step 10.2

Subtract $2$ from $1$ .

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

Step 11

Combine fractions.

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Step 11.1

Move the negative in front of the fraction.

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

Step 11.2

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

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

Step 11.3

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

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

Step 11.4

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

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

Step 12

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

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

Step 13

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

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

Step 14

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

$\frac{{(x - y)}^{\frac{1}{2}} - \frac{x}{2 {(x - y)}^{\frac{1}{2}}} (1 + 0)}{x - y}$

Step 15

Simplify the expression.

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Step 15.1

Add $1$ and $0$ .

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

Step 15.2

Multiply $- 1$ by $1$ .

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

Step 16

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

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

Step 17

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

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

Step 18

Combine the numerators over the common denominator.

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

Step 19

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

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Step 19.1

Move ${(x - y)}^{\frac{1}{2}}$ .

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

Step 19.2

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

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

Step 19.3

Combine the numerators over the common denominator.

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

Step 19.4

Add $1$ and $1$ .

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

Step 19.5

Divide $2$ by $2$ .

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

Step 20

Simplify ${(x - y)}^{1} \cdot 2$ .

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

Step 21

Move $2$ to the left of $x - y$ .

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

Step 22

Rewrite $\frac{\frac{2 (x - y) - x}{2 {(x - y)}^{\frac{1}{2}}}}{x - y}$ as a product.

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

Step 23

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

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

Step 24

Raise $x - y$ to the power of $1$ .

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

Step 25

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

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

Step 26

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

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

Step 27

Combine the numerators over the common denominator.

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

Step 28

Add $2$ and $1$ .

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

Step 29

Simplify.

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Step 29.1

Apply the distributive property.

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

Step 29.2

Simplify the numerator.

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Step 29.2.1

Multiply $- 1$ by $2$ .

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

Step 29.2.2

Subtract $x$ from $2 x$ .

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