Calculus Examples

$4 \sqrt{x y}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

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

$\frac{d}{d x} [4 {(x y)}^{\frac{1}{2}}]$

Step 1.2

Since $4$ is constant with respect to $x$ , the derivative of $4 {(x y)}^{\frac{1}{2}}$ with respect to $x$ is $4 \frac{d}{d x} [{(x y)}^{\frac{1}{2}}]$ .

$4 \frac{d}{d x} [{(x y)}^{\frac{1}{2}}]$

Step 2

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

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Step 2.1

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

$4 (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [x y])$

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $1$ .

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

Step 7

Move the negative in front of the fraction.

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

Step 8

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

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

Step 9

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

$4 (\frac{1}{2 {(x y)}^{\frac{1}{2}}} \frac{d}{d x} [x y])$

Step 10

Combine $\frac{1}{2 {(x y)}^{\frac{1}{2}}}$ and $4$ .

$\frac{4}{2 {(x y)}^{\frac{1}{2}}} \frac{d}{d x} [x y]$

Step 11

Factor $2$ out of $4$ .

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

Step 12

Cancel the common factors.

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Step 12.1

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

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

Step 12.2

Cancel the common factor.

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

Step 12.3

Rewrite the expression.

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

Step 13

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

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

Step 14

Combine fractions.

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Step 14.1

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

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

Step 14.2

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

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

Step 15

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

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

Step 16

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

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

Step 17

Simplify.

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Step 17.1

Apply the product rule to $x y$ .

$\frac{2 y}{x^{\frac{1}{2}} y^{\frac{1}{2}}}$

Step 17.2

Combine terms.

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Step 17.2.1

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

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

Step 17.2.2

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

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Step 17.2.2.1

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

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

Step 17.2.2.2

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

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Step 17.2.2.2.1

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

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

Step 17.2.2.2.2

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

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

Step 17.2.2.3

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

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

Step 17.2.2.4

Combine the numerators over the common denominator.

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

Step 17.2.2.5

Add $- 1$ and $2$ .

$\frac{2 y^{\frac{1}{2}}}{x^{\frac{1}{2}}}$