Find the Derivative - d/dy C=x square root of y^2+6

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

Step 1

Differentiate using the Constant Multiple Rule.

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

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

Since $x$ is constant with respect to $y$ , the derivative of $x {(y^{2} + 6)}^{\frac{1}{2}}$ with respect to $y$ is $x \frac{d}{d y} [{(y^{2} + 6)}^{\frac{1}{2}}]$ .

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

Step 2

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 $g (y) = y^{2} + 6$ .

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

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

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

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

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

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

Multiply $- 1$ by $2$ .

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

Subtract $2$ from $1$ .

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

Step 7

Combine fractions.

Move the negative in front of the fraction.

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

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

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

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

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

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Simplify terms.

Add $2 y$ and $0$ .

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

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Find the Derivative - d/dy C=x square root of y^2+6

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