Find dy/dx 12(x^2+y^2)^2=169xy^2

$12 {(x^{2} + y^{2})}^{2} = 169 x y^{2}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (12 {(x^{2} + y^{2})}^{2}) = \frac{d}{d x} (169 x y^{2})$

Step 2

Differentiate the left side of the equation.

Since $12$ is constant with respect to $x$ , the derivative of $12 {(x^{2} + y^{2})}^{2}$ with respect to $x$ is $12 \frac{d}{d x} [{(x^{2} + y^{2})}^{2}]$ .

$12 \frac{d}{d x} [{(x^{2} + y^{2})}^{2}]$

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

To apply the Chain Rule, set $u_{1}$ as $x^{2} + y^{2}$ .

$12 (\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [x^{2} + y^{2}])$

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

$12 (2 u_{1} \frac{d}{d x} [x^{2} + y^{2}])$

Replace all occurrences of $u_{1}$ with $x^{2} + y^{2}$ .

$12 (2 (x^{2} + y^{2}) \frac{d}{d x} [x^{2} + y^{2}])$

Differentiate.

Multiply $2$ by $12$ .

$24 ((x^{2} + y^{2}) \frac{d}{d x} [x^{2} + y^{2}])$

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

$24 (x^{2} + y^{2}) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [y^{2}])$

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

$24 (x^{2} + y^{2}) (2 x + \frac{d}{d x} [y^{2}])$

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

To apply the Chain Rule, set $u_{2}$ as $y$ .

$24 (x^{2} + y^{2}) (2 x + \frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [y])$

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

$24 (x^{2} + y^{2}) (2 x + 2 u_{2} \frac{d}{d x} [y])$

Replace all occurrences of $u_{2}$ with $y$ .

$24 (x^{2} + y^{2}) (2 x + 2 y \frac{d}{d x} [y])$

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$24 (x^{2} + y^{2}) (2 x + 2 y y')$

Apply the distributive property.

$(24 x^{2} + 24 y^{2}) (2 x + 2 y y')$

Step 3

Differentiate the right side of the equation.

Since $169$ is constant with respect to $x$ , the derivative of $169 x y^{2}$ with respect to $x$ is $169 \frac{d}{d x} [x y^{2}]$ .

$169 \frac{d}{d x} [x y^{2}]$

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 $g (x) = y^{2}$ .

$169 (x \frac{d}{d x} [y^{2}] + y^{2} \frac{d}{d x} [x])$

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

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

$169 (x (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x])$

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

$169 (x (2 u \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x])$

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

$169 (x (2 y \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x])$

Move $2$ to the left of $x$ .

$169 (2 \cdot x y \frac{d}{d x} [y] + y^{2} \frac{d}{d x} [x])$

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$169 (2 x y y' + y^{2} \frac{d}{d x} [x])$

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

$169 (2 x y y' + y^{2} \cdot 1)$

Multiply $y^{2}$ by $1$ .

$169 (2 x y y' + y^{2})$

Simplify.

Apply the distributive property.

$169 (2 x y y') + 169 y^{2}$

Multiply $2$ by $169$ .

$338 x y y' + 169 y^{2}$

Reorder terms.

$169 y^{2} + 338 x y y'$

Step 4

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

$(24 x^{2} + 24 y^{2}) (2 x + 2 y y') = 169 y^{2} + 338 x y y'$

Step 5

Solve for $y'$ .

Simplify $(24 x^{2} + 24 y^{2}) (2 x + 2 y y')$ .

Rewrite.

$0 + 0 + (24 x^{2} + 24 y^{2}) (2 x + 2 y y') = 169 y^{2} + 338 x y y'$

Simplify by adding zeros.

$(24 x^{2} + 24 y^{2}) (2 x + 2 y y') = 169 y^{2} + 338 x y y'$

Expand $(24 x^{2} + 24 y^{2}) (2 x + 2 y y')$ using the FOIL Method.

Apply the distributive property.

$24 x^{2} (2 x + 2 y y') + 24 y^{2} (2 x + 2 y y') = 169 y^{2} + 338 x y y'$

Apply the distributive property.

$24 x^{2} (2 x) + 24 x^{2} (2 y y') + 24 y^{2} (2 x + 2 y y') = 169 y^{2} + 338 x y y'$

Apply the distributive property.

$24 x^{2} (2 x) + 24 x^{2} (2 y y') + 24 y^{2} (2 x) + 24 y^{2} (2 y y') = 169 y^{2} + 338 x y y'$

Simplify each term.

Rewrite using the commutative property of multiplication.

$24 \cdot 2 x^{2} x + 24 x^{2} (2 y y') + 24 y^{2} (2 x) + 24 y^{2} (2 y y') = 169 y^{2} + 338 x y y'$

Multiply $x^{2}$ by $x$ by adding the exponents.

Move $x$ .

$24 \cdot 2 (x \cdot x^{2}) + 24 x^{2} (2 y y') + 24 y^{2} (2 x) + 24 y^{2} (2 y y') = 169 y^{2} + 338 x y y'$

Multiply $x$ by $x^{2}$ .

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

$24 \cdot 2 (x^{1} x^{2}) + 24 x^{2} (2 y y') + 24 y^{2} (2 x) + 24 y^{2} (2 y y') = 169 y^{2} + 338 x y y'$

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

$24 \cdot 2 x^{1 + 2} + 24 x^{2} (2 y y') + 24 y^{2} (2 x) + 24 y^{2} (2 y y') = 169 y^{2} + 338 x y y'$

Add $1$ and $2$ .

$24 \cdot 2 x^{3} + 24 x^{2} (2 y y') + 24 y^{2} (2 x) + 24 y^{2} (2 y y') = 169 y^{2} + 338 x y y'$

Multiply $24$ by $2$ .

$48 x^{3} + 24 x^{2} (2 y y') + 24 y^{2} (2 x) + 24 y^{2} (2 y y') = 169 y^{2} + 338 x y y'$

Multiply $2$ by $24$ .

$48 x^{3} + 48 x^{2} (y y') + 24 y^{2} (2 x) + 24 y^{2} (2 y y') = 169 y^{2} + 338 x y y'$

Rewrite using the commutative property of multiplication.

$48 x^{3} + 48 x^{2} y y' + 24 \cdot 2 y^{2} x + 24 y^{2} (2 y y') = 169 y^{2} + 338 x y y'$

Multiply $24$ by $2$ .

$48 x^{3} + 48 x^{2} y y' + 48 y^{2} x + 24 y^{2} (2 y y') = 169 y^{2} + 338 x y y'$

Multiply $y^{2}$ by $y$ by adding the exponents.

Move $y$ .

$48 x^{3} + 48 x^{2} y y' + 48 y^{2} x + 24 (y \cdot y^{2}) (2 y') = 169 y^{2} + 338 x y y'$

Multiply $y$ by $y^{2}$ .

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

$48 x^{3} + 48 x^{2} y y' + 48 y^{2} x + 24 (y^{1} y^{2}) (2 y') = 169 y^{2} + 338 x y y'$

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

$48 x^{3} + 48 x^{2} y y' + 48 y^{2} x + 24 y^{1 + 2} (2 y') = 169 y^{2} + 338 x y y'$

Add $1$ and $2$ .

$48 x^{3} + 48 x^{2} y y' + 48 y^{2} x + 24 y^{3} (2 y') = 169 y^{2} + 338 x y y'$

Rewrite using the commutative property of multiplication.

$48 x^{3} + 48 x^{2} y y' + 48 y^{2} x + 24 \cdot 2 y^{3} y' = 169 y^{2} + 338 x y y'$

Multiply $24$ by $2$ .

$48 x^{3} + 48 x^{2} y y' + 48 y^{2} x + 48 y^{3} y' = 169 y^{2} + 338 x y y'$

Subtract $338 x y y'$ from both sides of the equation.

$48 x^{3} + 48 x^{2} y y' + 48 y^{2} x + 48 y^{3} y' - 338 x y y' = 169 y^{2}$

Move all terms not containing $y'$ to the right side of the equation.

Subtract $48 x^{3}$ from both sides of the equation.

$48 x^{2} y y' + 48 y^{2} x + 48 y^{3} y' - 338 x y y' = 169 y^{2} - 48 x^{3}$

Subtract $48 y^{2} x$ from both sides of the equation.

$48 x^{2} y y' + 48 y^{3} y' - 338 x y y' = 169 y^{2} - 48 x^{3} - 48 y^{2} x$

Factor $2 y y'$ out of $48 x^{2} y y' + 48 y^{3} y' - 338 x y y'$ .

Factor $2 y y'$ out of $48 x^{2} y y'$ .

$2 y y' (24 x^{2}) + 48 y^{3} y' - 338 x y y' = 169 y^{2} - 48 x^{3} - 48 y^{2} x$

Factor $2 y y'$ out of $48 y^{3} y'$ .

$2 y y' (24 x^{2}) + 2 y y' (24 y^{2}) - 338 x y y' = 169 y^{2} - 48 x^{3} - 48 y^{2} x$

Factor $2 y y'$ out of $- 338 x y y'$ .

$2 y y' (24 x^{2}) + 2 y y' (24 y^{2}) + 2 y y' (- 169 x) = 169 y^{2} - 48 x^{3} - 48 y^{2} x$

Factor $2 y y'$ out of $2 y y' (24 x^{2}) + 2 y y' (24 y^{2})$ .

$2 y y' (24 x^{2} + 24 y^{2}) + 2 y y' (- 169 x) = 169 y^{2} - 48 x^{3} - 48 y^{2} x$

Factor $2 y y'$ out of $2 y y' (24 x^{2} + 24 y^{2}) + 2 y y' (- 169 x)$ .

$2 y y' (24 x^{2} + 24 y^{2} - 169 x) = 169 y^{2} - 48 x^{3} - 48 y^{2} x$

Divide each term in $2 y y' (24 x^{2} + 24 y^{2} - 169 x) = 169 y^{2} - 48 x^{3} - 48 y^{2} x$ by $2 y (24 x^{2} + 24 y^{2} - 169 x)$ and simplify.

Divide each term in $2 y y' (24 x^{2} + 24 y^{2} - 169 x) = 169 y^{2} - 48 x^{3} - 48 y^{2} x$ by $2 y (24 x^{2} + 24 y^{2} - 169 x)$ .

$\frac{2 y y' (24 x^{2} + 24 y^{2} - 169 x)}{2 y (24 x^{2} + 24 y^{2} - 169 x)} = \frac{169 y^{2}}{2 y (24 x^{2} + 24 y^{2} - 169 x)} + \frac{- 48 x^{3}}{2 y (24 x^{2} + 24 y^{2} - 169 x)} + \frac{- 48 y^{2} x}{2 y (24 x^{2} + 24 y^{2} - 169 x)}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

Rewrite the expression.

$\frac{y y' (24 x^{2} + 24 y^{2} - 169 x)}{y (24 x^{2} + 24 y^{2} - 169 x)} = \frac{169 y^{2}}{2 y (24 x^{2} + 24 y^{2} - 169 x)} + \frac{- 48 x^{3}}{2 y (24 x^{2} + 24 y^{2} - 169 x)} + \frac{- 48 y^{2} x}{2 y (24 x^{2} + 24 y^{2} - 169 x)}$

Cancel the common factor of $y$ .

Cancel the common factor.

Rewrite the expression.

$\frac{y' (24 x^{2} + 24 y^{2} - 169 x)}{24 x^{2} + 24 y^{2} - 169 x} = \frac{169 y^{2}}{2 y (24 x^{2} + 24 y^{2} - 169 x)} + \frac{- 48 x^{3}}{2 y (24 x^{2} + 24 y^{2} - 169 x)} + \frac{- 48 y^{2} x}{2 y (24 x^{2} + 24 y^{2} - 169 x)}$

Cancel the common factor of $24 x^{2} + 24 y^{2} - 169 x$ .

Cancel the common factor.

Divide $y'$ by $1$ .

$y' = \frac{169 y^{2}}{2 y (24 x^{2} + 24 y^{2} - 169 x)} + \frac{- 48 x^{3}}{2 y (24 x^{2} + 24 y^{2} - 169 x)} + \frac{- 48 y^{2} x}{2 y (24 x^{2} + 24 y^{2} - 169 x)}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $y^{2}$ and $y$ .

Factor $y$ out of $169 y^{2}$ .

$y' = \frac{y (169 y)}{2 y (24 x^{2} + 24 y^{2} - 169 x)} + \frac{- 48 x^{3}}{2 y (24 x^{2} + 24 y^{2} - 169 x)} + \frac{- 48 y^{2} x}{2 y (24 x^{2} + 24 y^{2} - 169 x)}$

Cancel the common factors.

Factor $y$ out of $2 y (24 x^{2} + 24 y^{2} - 169 x)$ .

$y' = \frac{y (169 y)}{y (2 (24 x^{2} + 24 y^{2} - 169 x))} + \frac{- 48 x^{3}}{2 y (24 x^{2} + 24 y^{2} - 169 x)} + \frac{- 48 y^{2} x}{2 y (24 x^{2} + 24 y^{2} - 169 x)}$

Cancel the common factor.

$y' = \frac{y (169 y)}{y (2 (24 x^{2} + 24 y^{2} - 169 x))} + \frac{- 48 x^{3}}{2 y (24 x^{2} + 24 y^{2} - 169 x)} + \frac{- 48 y^{2} x}{2 y (24 x^{2} + 24 y^{2} - 169 x)}$

Rewrite the expression.

$y' = \frac{169 y}{2 (24 x^{2} + 24 y^{2} - 169 x)} + \frac{- 48 x^{3}}{2 y (24 x^{2} + 24 y^{2} - 169 x)} + \frac{- 48 y^{2} x}{2 y (24 x^{2} + 24 y^{2} - 169 x)}$

Cancel the common factor of $- 48$ and $2$ .

Factor $2$ out of $- 48 x^{3}$ .

$y' = \frac{169 y}{2 (24 x^{2} + 24 y^{2} - 169 x)} + \frac{2 (- 24 x^{3})}{2 y (24 x^{2} + 24 y^{2} - 169 x)} + \frac{- 48 y^{2} x}{2 y (24 x^{2} + 24 y^{2} - 169 x)}$

Cancel the common factors.

Factor $2$ out of $2 y (24 x^{2} + 24 y^{2} - 169 x)$ .

$y' = \frac{169 y}{2 (24 x^{2} + 24 y^{2} - 169 x)} + \frac{2 (- 24 x^{3})}{2 (y (24 x^{2} + 24 y^{2} - 169 x))} + \frac{- 48 y^{2} x}{2 y (24 x^{2} + 24 y^{2} - 169 x)}$

Cancel the common factor.

$y' = \frac{169 y}{2 (24 x^{2} + 24 y^{2} - 169 x)} + \frac{2 (- 24 x^{3})}{2 (y (24 x^{2} + 24 y^{2} - 169 x))} + \frac{- 48 y^{2} x}{2 y (24 x^{2} + 24 y^{2} - 169 x)}$

Rewrite the expression.

$y' = \frac{169 y}{2 (24 x^{2} + 24 y^{2} - 169 x)} + \frac{- 24 x^{3}}{y (24 x^{2} + 24 y^{2} - 169 x)} + \frac{- 48 y^{2} x}{2 y (24 x^{2} + 24 y^{2} - 169 x)}$

Move the negative in front of the fraction.

$y' = \frac{169 y}{2 (24 x^{2} + 24 y^{2} - 169 x)} - \frac{(24) x^{3}}{y (24 x^{2} + 24 y^{2} - 169 x)} + \frac{- 48 y^{2} x}{2 y (24 x^{2} + 24 y^{2} - 169 x)}$

Cancel the common factor of $- 48$ and $2$ .

Factor $2$ out of $- 48 y^{2} x$ .

$y' = \frac{169 y}{2 (24 x^{2} + 24 y^{2} - 169 x)} - \frac{24 x^{3}}{y (24 x^{2} + 24 y^{2} - 169 x)} + \frac{2 (- 24 y^{2} x)}{2 y (24 x^{2} + 24 y^{2} - 169 x)}$

Cancel the common factors.

Factor $2$ out of $2 y (24 x^{2} + 24 y^{2} - 169 x)$ .

$y' = \frac{169 y}{2 (24 x^{2} + 24 y^{2} - 169 x)} - \frac{24 x^{3}}{y (24 x^{2} + 24 y^{2} - 169 x)} + \frac{2 (- 24 y^{2} x)}{2 (y (24 x^{2} + 24 y^{2} - 169 x))}$

Cancel the common factor.

$y' = \frac{169 y}{2 (24 x^{2} + 24 y^{2} - 169 x)} - \frac{24 x^{3}}{y (24 x^{2} + 24 y^{2} - 169 x)} + \frac{2 (- 24 y^{2} x)}{2 (y (24 x^{2} + 24 y^{2} - 169 x))}$

Rewrite the expression.

$y' = \frac{169 y}{2 (24 x^{2} + 24 y^{2} - 169 x)} - \frac{24 x^{3}}{y (24 x^{2} + 24 y^{2} - 169 x)} + \frac{- 24 y^{2} x}{y (24 x^{2} + 24 y^{2} - 169 x)}$

Cancel the common factor of $y^{2}$ and $y$ .

Factor $y$ out of $- 24 y^{2} x$ .

$y' = \frac{169 y}{2 (24 x^{2} + 24 y^{2} - 169 x)} - \frac{24 x^{3}}{y (24 x^{2} + 24 y^{2} - 169 x)} + \frac{y (- 24 y x)}{y (24 x^{2} + 24 y^{2} - 169 x)}$

Cancel the common factors.

Cancel the common factor.

$y' = \frac{169 y}{2 (24 x^{2} + 24 y^{2} - 169 x)} - \frac{24 x^{3}}{y (24 x^{2} + 24 y^{2} - 169 x)} + \frac{y (- 24 y x)}{y (24 x^{2} + 24 y^{2} - 169 x)}$

Rewrite the expression.

$y' = \frac{169 y}{2 (24 x^{2} + 24 y^{2} - 169 x)} - \frac{24 x^{3}}{y (24 x^{2} + 24 y^{2} - 169 x)} + \frac{- 24 y x}{24 x^{2} + 24 y^{2} - 169 x}$

Move the negative in front of the fraction.

$y' = \frac{169 y}{2 (24 x^{2} + 24 y^{2} - 169 x)} - \frac{24 x^{3}}{y (24 x^{2} + 24 y^{2} - 169 x)} - \frac{24 y x}{24 x^{2} + 24 y^{2} - 169 x}$

To write $- \frac{24 y x}{24 x^{2} + 24 y^{2} - 169 x}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y' = - \frac{24 x^{3}}{y (24 x^{2} + 24 y^{2} - 169 x)} + \frac{169 y}{2 (24 x^{2} + 24 y^{2} - 169 x)} - \frac{24 y x}{24 x^{2} + 24 y^{2} - 169 x} \cdot \frac{2}{2}$

Write each expression with a common denominator of $2 (24 x^{2} + 24 y^{2} - 169 x)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{24 y x}{24 x^{2} + 24 y^{2} - 169 x}$ by $\frac{2}{2}$ .

$y' = - \frac{24 x^{3}}{y (24 x^{2} + 24 y^{2} - 169 x)} + \frac{169 y}{2 (24 x^{2} + 24 y^{2} - 169 x)} - \frac{24 y x \cdot 2}{(24 x^{2} + 24 y^{2} - 169 x) \cdot 2}$

Reorder the factors of $(24 x^{2} + 24 y^{2} - 169 x) \cdot 2$ .

$y' = - \frac{24 x^{3}}{y (24 x^{2} + 24 y^{2} - 169 x)} + \frac{169 y}{2 (24 x^{2} + 24 y^{2} - 169 x)} - \frac{24 y x \cdot 2}{2 (24 x^{2} + 24 y^{2} - 169 x)}$

Combine the numerators over the common denominator.

$y' = - \frac{24 x^{3}}{y (24 x^{2} + 24 y^{2} - 169 x)} + \frac{169 y - 24 y x \cdot 2}{2 (24 x^{2} + 24 y^{2} - 169 x)}$

Simplify the numerator.

Factor $y$ out of $169 y - 24 y x \cdot 2$ .

Factor $y$ out of $169 y$ .

$y' = - \frac{24 x^{3}}{y (24 x^{2} + 24 y^{2} - 169 x)} + \frac{y \cdot 169 - 24 y x \cdot 2}{2 (24 x^{2} + 24 y^{2} - 169 x)}$

Factor $y$ out of $- 24 y x \cdot 2$ .

$y' = - \frac{24 x^{3}}{y (24 x^{2} + 24 y^{2} - 169 x)} + \frac{y \cdot 169 + y (- 24 x \cdot 2)}{2 (24 x^{2} + 24 y^{2} - 169 x)}$

Factor $y$ out of $y \cdot 169 + y (- 24 x \cdot 2)$ .

$y' = - \frac{24 x^{3}}{y (24 x^{2} + 24 y^{2} - 169 x)} + \frac{y (169 - 24 x \cdot 2)}{2 (24 x^{2} + 24 y^{2} - 169 x)}$

Multiply $2$ by $- 24$ .

$y' = - \frac{24 x^{3}}{y (24 x^{2} + 24 y^{2} - 169 x)} + \frac{y (169 - 48 x)}{2 (24 x^{2} + 24 y^{2} - 169 x)}$

To write $- \frac{24 x^{3}}{y (24 x^{2} + 24 y^{2} - 169 x)}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y' = - \frac{24 x^{3}}{y (24 x^{2} + 24 y^{2} - 169 x)} \cdot \frac{2}{2} + \frac{y (169 - 48 x)}{2 (24 x^{2} + 24 y^{2} - 169 x)}$

To write $\frac{y (169 - 48 x)}{2 (24 x^{2} + 24 y^{2} - 169 x)}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$y' = - \frac{24 x^{3}}{y (24 x^{2} + 24 y^{2} - 169 x)} \cdot \frac{2}{2} + \frac{y (169 - 48 x)}{2 (24 x^{2} + 24 y^{2} - 169 x)} \cdot \frac{y}{y}$

Write each expression with a common denominator of $y (24 x^{2} + 24 y^{2} - 169 x) \cdot 2$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{24 x^{3}}{y (24 x^{2} + 24 y^{2} - 169 x)}$ by $\frac{2}{2}$ .

$y' = - \frac{24 x^{3} \cdot 2}{y (24 x^{2} + 24 y^{2} - 169 x) \cdot 2} + \frac{y (169 - 48 x)}{2 (24 x^{2} + 24 y^{2} - 169 x)} \cdot \frac{y}{y}$

Multiply $\frac{y (169 - 48 x)}{2 (24 x^{2} + 24 y^{2} - 169 x)}$ by $\frac{y}{y}$ .

$y' = - \frac{24 x^{3} \cdot 2}{y (24 x^{2} + 24 y^{2} - 169 x) \cdot 2} + \frac{y (169 - 48 x) y}{2 (24 x^{2} + 24 y^{2} - 169 x) y}$

Reorder the factors of $y (24 x^{2} + 24 y^{2} - 169 x) \cdot 2$ .

$y' = - \frac{24 x^{3} \cdot 2}{2 y (24 x^{2} + 24 y^{2} - 169 x)} + \frac{y (169 - 48 x) y}{2 (24 x^{2} + 24 y^{2} - 169 x) y}$

Reorder the factors of $2 (24 x^{2} + 24 y^{2} - 169 x) y$ .

$y' = - \frac{24 x^{3} \cdot 2}{2 y (24 x^{2} + 24 y^{2} - 169 x)} + \frac{y (169 - 48 x) y}{2 y (24 x^{2} + 24 y^{2} - 169 x)}$

Combine the numerators over the common denominator.

$y' = \frac{- 24 x^{3} \cdot 2 + y (169 - 48 x) y}{2 y (24 x^{2} + 24 y^{2} - 169 x)}$

Simplify the numerator.

Multiply $2$ by $- 24$ .

$y' = \frac{- 48 x^{3} + y (169 - 48 x) y}{2 y (24 x^{2} + 24 y^{2} - 169 x)}$

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

$y' = \frac{- 48 x^{3} + y \cdot y (169 - 48 x)}{2 y (24 x^{2} + 24 y^{2} - 169 x)}$

Multiply $y$ by $y$ .

$y' = \frac{- 48 x^{3} + y^{2} (169 - 48 x)}{2 y (24 x^{2} + 24 y^{2} - 169 x)}$

Apply the distributive property.

$y' = \frac{- 48 x^{3} + y^{2} \cdot 169 + y^{2} (- 48 x)}{2 y (24 x^{2} + 24 y^{2} - 169 x)}$

Move $169$ to the left of $y^{2}$ .

$y' = \frac{- 48 x^{3} + 169 \cdot y^{2} + y^{2} (- 48 x)}{2 y (24 x^{2} + 24 y^{2} - 169 x)}$

Rewrite using the commutative property of multiplication.

$y' = \frac{- 48 x^{3} + 169 y^{2} - 48 y^{2} x}{2 y (24 x^{2} + 24 y^{2} - 169 x)}$

Simplify with factoring out.

Factor $- 1$ out of $- 48 x^{3}$ .

$y' = \frac{- (48 x^{3}) + 169 y^{2} - 48 y^{2} x}{2 y (24 x^{2} + 24 y^{2} - 169 x)}$

Factor $- 1$ out of $169 y^{2}$ .

$y' = \frac{- (48 x^{3}) - (- 169 y^{2}) - 48 y^{2} x}{2 y (24 x^{2} + 24 y^{2} - 169 x)}$

Factor $- 1$ out of $- (48 x^{3}) - (- 169 y^{2})$ .

$y' = \frac{- (48 x^{3} - 169 y^{2}) - 48 y^{2} x}{2 y (24 x^{2} + 24 y^{2} - 169 x)}$

Factor $- 1$ out of $- 48 y^{2} x$ .

$y' = \frac{- (48 x^{3} - 169 y^{2}) - (48 y^{2} x)}{2 y (24 x^{2} + 24 y^{2} - 169 x)}$

Factor $- 1$ out of $- (48 x^{3} - 169 y^{2}) - (48 y^{2} x)$ .

$y' = \frac{- (48 x^{3} - 169 y^{2} + 48 y^{2} x)}{2 y (24 x^{2} + 24 y^{2} - 169 x)}$

Simplify the expression.

Rewrite $- (48 x^{3} - 169 y^{2} + 48 y^{2} x)$ as $- 1 (48 x^{3} - 169 y^{2} + 48 y^{2} x)$ .

$y' = \frac{- 1 (48 x^{3} - 169 y^{2} + 48 y^{2} x)}{2 y (24 x^{2} + 24 y^{2} - 169 x)}$

Move the negative in front of the fraction.

$y' = - \frac{48 x^{3} - 169 y^{2} + 48 y^{2} x}{2 y (24 x^{2} + 24 y^{2} - 169 x)}$

Step 6

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

$\frac{d y}{d x} = - \frac{48 x^{3} - 169 y^{2} + 48 y^{2} x}{2 y (24 x^{2} + 24 y^{2} - 169 x)}$

Find dy/dx 12(x^2+y^2)^2=169xy^2

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