Find the Derivative – d/dx square root of xy

$\sqrt{x y}$

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

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

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

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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Multiply $- 1$ by $2$ .

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

Subtract $2$ from $1$ .

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

Combine fractions.

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Move the negative in front of the fraction.

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

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

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

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

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

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

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

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

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

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

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

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

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

Simplify.

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Apply the product rule to $x y$ .

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

Combine terms.

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

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Subtract $1$ from $2$ .

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

Find the Derivative – d/dx square root of xy

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