Find ds/dy s = square root of 2x^2+6y^2

$s = \sqrt{2 x^{2} + 6 y^{2}}$

Step 1

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

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

Step 2

Differentiate both sides of the equation.

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

Step 3

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

$s'$

Step 4

Differentiate the right side of the equation.

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{\frac{1}{2}}$ and $g (y) = 2 x^{2} + 6 y^{2}$ .

To apply the Chain Rule, set $u$ as $2 x^{2} + 6 y^{2}$ .

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

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

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

Replace all occurrences of $u$ with $2 x^{2} + 6 y^{2}$ .

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 1$ by $2$ .

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

Subtract $2$ from $1$ .

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

Combine fractions.

Move the negative in front of the fraction.

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

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

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

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

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

By the Sum Rule, the derivative of $2 x^{2} + 6 y^{2}$ with respect to $y$ is $\frac{d}{d y} [2 x^{2}] + \frac{d}{d y} [6 y^{2}]$ .

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

Since $2 x^{2}$ is constant with respect to $y$ , the derivative of $2 x^{2}$ with respect to $y$ is $0$ .

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

Add $0$ and $\frac{d}{d y} [6 y^{2}]$ .

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

Since $6$ is constant with respect to $y$ , the derivative of $6 y^{2}$ with respect to $y$ is $6 \frac{d}{d y} [y^{2}]$ .

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

Simplify terms.

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

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

Factor $2$ out of $6$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Combine fractions.

Combine $2$ and $\frac{3}{{(2 x^{2} + 6 y^{2})}^{\frac{1}{2}}}$ .

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

Multiply $2$ by $3$ .

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

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

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

Step 5

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

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

Step 6

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

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

Find ds/dy s = square root of 2x^2+6y^2

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