Find dy/dx y = square root of x(x+6)

$y = \sqrt{x (x + 6)}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x (x + 6)}$ as ${(x (x + 6))}^{\frac{1}{2}}$ .

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

Differentiate both sides of the equation.

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

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

$y'$

Differentiate the right side of the equation.

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 $g (x) = x (x + 6)$ .

To apply the Chain Rule, set $u$ as $x (x + 6)$ .

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [x (x + 6)]$

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

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [x (x + 6)]$

Replace all occurrences of $u$ with $x (x + 6)$ .

$\frac{1}{2} {(x (x + 6))}^{\frac{1}{2} - 1} \frac{d}{d x} [x (x + 6)]$

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

$\frac{1}{2} {(x (x + 6))}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [x (x + 6)]$

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

$\frac{1}{2} {(x (x + 6))}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [x (x + 6)]$

Combine the numerators over the common denominator.

$\frac{1}{2} {(x (x + 6))}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [x (x + 6)]$

Simplify the numerator.

Multiply $- 1$ by $2$ .

$\frac{1}{2} {(x (x + 6))}^{\frac{1 - 2}{2}} \frac{d}{d x} [x (x + 6)]$

Subtract $2$ from $1$ .

$\frac{1}{2} {(x (x + 6))}^{\frac{- 1}{2}} \frac{d}{d x} [x (x + 6)]$

Combine fractions.

Move the negative in front of the fraction.

$\frac{1}{2} {(x (x + 6))}^{- \frac{1}{2}} \frac{d}{d x} [x (x + 6)]$

Combine $\frac{1}{2}$ and ${(x (x + 6))}^{- \frac{1}{2}}$ .

$\frac{{(x (x + 6))}^{- \frac{1}{2}}}{2} \frac{d}{d x} [x (x + 6)]$

Move ${(x (x + 6))}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2 {(x (x + 6))}^{\frac{1}{2}}} \frac{d}{d x} [x (x + 6)]$

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 $g (x) = x + 6$ .

$\frac{1}{2 {(x (x + 6))}^{\frac{1}{2}}} (x \frac{d}{d x} [x + 6] + (x + 6) \frac{d}{d x} [x])$

Differentiate.

By the Sum Rule, the derivative of $x + 6$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [6]$ .

$\frac{1}{2 {(x (x + 6))}^{\frac{1}{2}}} (x (\frac{d}{d x} [x] + \frac{d}{d x} [6]) + (x + 6) \frac{d}{d x} [x])$

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

$\frac{1}{2 {(x (x + 6))}^{\frac{1}{2}}} (x (1 + \frac{d}{d x} [6]) + (x + 6) \frac{d}{d x} [x])$

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$\frac{1}{2 {(x (x + 6))}^{\frac{1}{2}}} (x (1 + 0) + (x + 6) \frac{d}{d x} [x])$

Simplify the expression.

Add $1$ and $0$ .

$\frac{1}{2 {(x (x + 6))}^{\frac{1}{2}}} (x \cdot 1 + (x + 6) \frac{d}{d x} [x])$

Multiply $x$ by $1$ .

$\frac{1}{2 {(x (x + 6))}^{\frac{1}{2}}} (x + (x + 6) \frac{d}{d x} [x])$

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

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

Simplify by adding terms.

Multiply $x + 6$ by $1$ .

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

Add $x$ and $x$ .

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

Simplify.

Apply the product rule to $x (x + 6)$ .

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

Reorder the factors of $\frac{1}{2 x^{\frac{1}{2}} {(x + 6)}^{\frac{1}{2}}} (2 x + 6)$ .

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

Multiply $2 x + 6$ by $\frac{1}{2 x^{\frac{1}{2}} {(x + 6)}^{\frac{1}{2}}}$ .

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

Factor $2$ out of $2 x$ .

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

Factor $2$ out of $6$ .

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

Factor $2$ out of $2 (x) + 2 (3)$ .

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

Cancel the common factors.

Factor $2$ out of $2 x^{\frac{1}{2}} {(x + 6)}^{\frac{1}{2}}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

Find dy/dx y = square root of x(x+6)

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