Calculus Examples

$D x (\sqrt{5 x y})$

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 $\sqrt{5 x y}$ as ${(5 x y)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [D x {(5 x y)}^{\frac{1}{2}}]$

Step 1.2

Simplify with factoring out.

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Step 1.2.1

Factor $5$ out of $5 x y$ .

$\frac{d}{d x} [D x {(5 (x y))}^{\frac{1}{2}}]$

Step 1.2.2

Apply the product rule to $5 (x y)$ .

$\frac{d}{d x} [D x (5^{\frac{1}{2}} {(x y)}^{\frac{1}{2}})]$

Step 1.3

Since $D \cdot 5^{\frac{1}{2}}$ is constant with respect to $x$ , the derivative of $D x (5^{\frac{1}{2}} {(x y)}^{\frac{1}{2}})$ with respect to $x$ is $D \cdot 5^{\frac{1}{2}} \frac{d}{d x} [x ({(x y)}^{\frac{1}{2}})]$ .

$D \cdot 5^{\frac{1}{2}} \frac{d}{d x} [x ({(x y)}^{\frac{1}{2}})]$

Step 2

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

$D \cdot 5^{\frac{1}{2}} (x \frac{d}{d x} [{(x y)}^{\frac{1}{2}}] + {(x y)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 3

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 3.1

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

$D \cdot 5^{\frac{1}{2}} (x (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [x y]) + {(x y)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 3.2

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

$D \cdot 5^{\frac{1}{2}} (x (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [x y]) + {(x y)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 3.3

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

$D \cdot 5^{\frac{1}{2}} (x (\frac{1}{2} {(x y)}^{\frac{1}{2} - 1} \frac{d}{d x} [x y]) + {(x y)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 4

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

$D \cdot 5^{\frac{1}{2}} (x (\frac{1}{2} {(x y)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [x y]) + {(x y)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 5

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

$D \cdot 5^{\frac{1}{2}} (x (\frac{1}{2} {(x y)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [x y]) + {(x y)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 6

Combine the numerators over the common denominator.

$D \cdot 5^{\frac{1}{2}} (x (\frac{1}{2} {(x y)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [x y]) + {(x y)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

$D \cdot 5^{\frac{1}{2}} (x (\frac{1}{2} {(x y)}^{\frac{1 - 2}{2}} \frac{d}{d x} [x y]) + {(x y)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 7.2

Subtract $2$ from $1$ .

$D \cdot 5^{\frac{1}{2}} (x (\frac{1}{2} {(x y)}^{\frac{- 1}{2}} \frac{d}{d x} [x y]) + {(x y)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

$D \cdot 5^{\frac{1}{2}} (x (\frac{1}{2} {(x y)}^{- \frac{1}{2}} \frac{d}{d x} [x y]) + {(x y)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 8.2

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

$D \cdot 5^{\frac{1}{2}} (x (\frac{{(x y)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [x y]) + {(x y)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 8.3

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

$D \cdot 5^{\frac{1}{2}} (x (\frac{1}{2 {(x y)}^{\frac{1}{2}}} \frac{d}{d x} [x y]) + {(x y)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 8.4

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

$D \cdot 5^{\frac{1}{2}} (\frac{x}{2 {(x y)}^{\frac{1}{2}}} \frac{d}{d x} [x y] + {(x y)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 9

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

$D \cdot 5^{\frac{1}{2}} (\frac{x}{2 {(x y)}^{\frac{1}{2}}} (y \frac{d}{d x} [x]) + {(x y)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 10

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

$D \cdot 5^{\frac{1}{2}} (\frac{y x}{2 {(x y)}^{\frac{1}{2}}} \frac{d}{d x} [x] + {(x y)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 11

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

$D \cdot 5^{\frac{1}{2}} (\frac{y x}{2 {(x y)}^{\frac{1}{2}}} \cdot 1 + {(x y)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 12

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

$D \cdot 5^{\frac{1}{2}} (\frac{y x}{2 {(x y)}^{\frac{1}{2}}} + {(x y)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 13

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

$D \cdot 5^{\frac{1}{2}} (\frac{y x}{2 {(x y)}^{\frac{1}{2}}} + {(x y)}^{\frac{1}{2}} \cdot 1)$

Step 14

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

$D \cdot 5^{\frac{1}{2}} (\frac{y x}{2 {(x y)}^{\frac{1}{2}}} + {(x y)}^{\frac{1}{2}})$

Step 15

Simplify.

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Step 15.1

Apply the product rule to $x y$ .

$D \cdot 5^{\frac{1}{2}} (\frac{y x}{2 (x^{\frac{1}{2}} y^{\frac{1}{2}})} + {(x y)}^{\frac{1}{2}})$

Step 15.2

Apply the product rule to $x y$ .

$D \cdot 5^{\frac{1}{2}} (\frac{y x}{2 (x^{\frac{1}{2}} y^{\frac{1}{2}})} + x^{\frac{1}{2}} y^{\frac{1}{2}})$

Step 15.3

Apply the distributive property.

$D \cdot 5^{\frac{1}{2}} \frac{y x}{2 (x^{\frac{1}{2}} y^{\frac{1}{2}})} + D \cdot 5^{\frac{1}{2}} (x^{\frac{1}{2}} y^{\frac{1}{2}})$

Step 15.4

Combine terms.

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Step 15.4.1

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

$D \cdot 5^{\frac{1}{2}} \frac{y x y^{- \frac{1}{2}}}{2 x^{\frac{1}{2}}} + D \cdot 5^{\frac{1}{2}} (x^{\frac{1}{2}} y^{\frac{1}{2}})$

Step 15.4.2

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

$D \cdot 5^{\frac{1}{2}} \frac{y x y^{- \frac{1}{2}} x^{- \frac{1}{2}}}{2} + D \cdot 5^{\frac{1}{2}} (x^{\frac{1}{2}} y^{\frac{1}{2}})$

Step 15.4.3

Simplify the numerator.

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Step 15.4.3.1

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

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Step 15.4.3.1.1

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

$D \cdot 5^{\frac{1}{2}} \frac{y^{- \frac{1}{2}} y x \cdot x^{- \frac{1}{2}}}{2} + D \cdot 5^{\frac{1}{2}} (x^{\frac{1}{2}} y^{\frac{1}{2}})$

Step 15.4.3.1.2

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

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Step 15.4.3.1.2.1

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

$D \cdot 5^{\frac{1}{2}} \frac{y^{- \frac{1}{2}} y^{1} x \cdot x^{- \frac{1}{2}}}{2} + D \cdot 5^{\frac{1}{2}} (x^{\frac{1}{2}} y^{\frac{1}{2}})$

Step 15.4.3.1.2.2

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

$D \cdot 5^{\frac{1}{2}} \frac{y^{- \frac{1}{2} + 1} x \cdot x^{- \frac{1}{2}}}{2} + D \cdot 5^{\frac{1}{2}} (x^{\frac{1}{2}} y^{\frac{1}{2}})$

Step 15.4.3.1.3

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

$D \cdot 5^{\frac{1}{2}} \frac{y^{- \frac{1}{2} + \frac{2}{2}} x \cdot x^{- \frac{1}{2}}}{2} + D \cdot 5^{\frac{1}{2}} (x^{\frac{1}{2}} y^{\frac{1}{2}})$

Step 15.4.3.1.4

Combine the numerators over the common denominator.

$D \cdot 5^{\frac{1}{2}} \frac{y^{\frac{- 1 + 2}{2}} x \cdot x^{- \frac{1}{2}}}{2} + D \cdot 5^{\frac{1}{2}} (x^{\frac{1}{2}} y^{\frac{1}{2}})$

Step 15.4.3.1.5

Add $- 1$ and $2$ .

$D \cdot 5^{\frac{1}{2}} \frac{y^{\frac{1}{2}} x \cdot x^{- \frac{1}{2}}}{2} + D \cdot 5^{\frac{1}{2}} (x^{\frac{1}{2}} y^{\frac{1}{2}})$

Step 15.4.3.2

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

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Step 15.4.3.2.1

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

$D \cdot 5^{\frac{1}{2}} \frac{y^{\frac{1}{2}} (x^{- \frac{1}{2}} x)}{2} + D \cdot 5^{\frac{1}{2}} (x^{\frac{1}{2}} y^{\frac{1}{2}})$

Step 15.4.3.2.2

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

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Step 15.4.3.2.2.1

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

$D \cdot 5^{\frac{1}{2}} \frac{y^{\frac{1}{2}} (x^{- \frac{1}{2}} x^{1})}{2} + D \cdot 5^{\frac{1}{2}} (x^{\frac{1}{2}} y^{\frac{1}{2}})$

Step 15.4.3.2.2.2

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

$D \cdot 5^{\frac{1}{2}} \frac{y^{\frac{1}{2}} x^{- \frac{1}{2} + 1}}{2} + D \cdot 5^{\frac{1}{2}} (x^{\frac{1}{2}} y^{\frac{1}{2}})$

Step 15.4.3.2.3

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

$D \cdot 5^{\frac{1}{2}} \frac{y^{\frac{1}{2}} x^{- \frac{1}{2} + \frac{2}{2}}}{2} + D \cdot 5^{\frac{1}{2}} (x^{\frac{1}{2}} y^{\frac{1}{2}})$

Step 15.4.3.2.4

Combine the numerators over the common denominator.

$D \cdot 5^{\frac{1}{2}} \frac{y^{\frac{1}{2}} x^{\frac{- 1 + 2}{2}}}{2} + D \cdot 5^{\frac{1}{2}} (x^{\frac{1}{2}} y^{\frac{1}{2}})$

Step 15.4.3.2.5

Add $- 1$ and $2$ .

$D \cdot 5^{\frac{1}{2}} \frac{y^{\frac{1}{2}} x^{\frac{1}{2}}}{2} + D \cdot 5^{\frac{1}{2}} (x^{\frac{1}{2}} y^{\frac{1}{2}})$

Step 15.4.4

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

$5^{\frac{1}{2}} \frac{D (y^{\frac{1}{2}} x^{\frac{1}{2}})}{2} + D \cdot 5^{\frac{1}{2}} (x^{\frac{1}{2}} y^{\frac{1}{2}})$

Step 15.4.5

Combine $5^{\frac{1}{2}}$ and $\frac{D (y^{\frac{1}{2}} x^{\frac{1}{2}})}{2}$ .

$\frac{5^{\frac{1}{2}} (D (y^{\frac{1}{2}} x^{\frac{1}{2}}))}{2} + D \cdot 5^{\frac{1}{2}} (x^{\frac{1}{2}} y^{\frac{1}{2}})$

Step 15.4.6

Reorder $D$ and $5^{\frac{1}{2}}$ .

$\frac{5^{\frac{1}{2}} D x^{\frac{1}{2}} y^{\frac{1}{2}}}{2} + 5^{\frac{1}{2}} \cdot D x^{\frac{1}{2}} y^{\frac{1}{2}}$

Step 15.4.7

To write $5^{\frac{1}{2}} D x^{\frac{1}{2}} y^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{5^{\frac{1}{2}} D x^{\frac{1}{2}} y^{\frac{1}{2}}}{2} + 5^{\frac{1}{2}} D x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot \frac{2}{2}$

Step 15.4.8

Combine $5^{\frac{1}{2}} D x^{\frac{1}{2}} y^{\frac{1}{2}}$ and $\frac{2}{2}$ .

$\frac{5^{\frac{1}{2}} D x^{\frac{1}{2}} y^{\frac{1}{2}}}{2} + \frac{5^{\frac{1}{2}} D x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 2}{2}$

Step 15.4.9

Combine the numerators over the common denominator.

$\frac{5^{\frac{1}{2}} D x^{\frac{1}{2}} y^{\frac{1}{2}} + 5^{\frac{1}{2}} D x^{\frac{1}{2}} y^{\frac{1}{2}} \cdot 2}{2}$

Step 15.4.10

Move $2$ to the left of $5^{\frac{1}{2}} D x^{\frac{1}{2}} y^{\frac{1}{2}}$ .

$\frac{5^{\frac{1}{2}} D x^{\frac{1}{2}} y^{\frac{1}{2}} + 2 \cdot (5^{\frac{1}{2}} D x^{\frac{1}{2}} y^{\frac{1}{2}})}{2}$

Step 15.4.11

Add $5^{\frac{1}{2}} D x^{\frac{1}{2}} y^{\frac{1}{2}}$ and $2 \cdot 5^{\frac{1}{2}} D x^{\frac{1}{2}} y^{\frac{1}{2}}$ .

$\frac{3 \cdot 5^{\frac{1}{2}} D x^{\frac{1}{2}} y^{\frac{1}{2}}}{2}$

Step 15.5

Reorder terms.

$\frac{3 \cdot 5^{\frac{1}{2}} x^{\frac{1}{2}} y^{\frac{1}{2}} D}{2}$

Step 15.6

Reorder factors in $\frac{3 \cdot 5^{\frac{1}{2}} x^{\frac{1}{2}} y^{\frac{1}{2}} D}{2}$ .

$\frac{3 D \cdot 5^{\frac{1}{2}} x^{\frac{1}{2}} y^{\frac{1}{2}}}{2}$