Find dy/dx 5(x^2+y^2)^2=169(x^2-y^2)

$5 {(x^{2} + y^{2})}^{2} = 169 (x^{2} - y^{2})$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (5 {(x^{2} + y^{2})}^{2}) = \frac{d}{d x} (169 (x^{2} - y^{2}))$

Step 2

Differentiate the left side of the equation.

Since $5$ is constant with respect to $x$ , the derivative of $5 {(x^{2} + y^{2})}^{2}$ with respect to $x$ is $5 \frac{d}{d x} [{(x^{2} + y^{2})}^{2}]$ .

$5 \frac{d}{d x} [{(x^{2} + y^{2})}^{2}]$

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 $x^{2} + y^{2}$ .

$5 (\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [x^{2} + y^{2}])$

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

$5 (2 u_{1} \frac{d}{d x} [x^{2} + y^{2}])$

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

$5 (2 (x^{2} + y^{2}) \frac{d}{d x} [x^{2} + y^{2}])$

Differentiate.

Multiply $2$ by $5$ .

$10 ((x^{2} + y^{2}) \frac{d}{d x} [x^{2} + y^{2}])$

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

$10 (x^{2} + y^{2}) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [y^{2}])$

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

$10 (x^{2} + y^{2}) (2 x + \frac{d}{d x} [y^{2}])$

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 $y$ .

$10 (x^{2} + y^{2}) (2 x + \frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [y])$

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

$10 (x^{2} + y^{2}) (2 x + 2 u_{2} \frac{d}{d x} [y])$

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

$10 (x^{2} + y^{2}) (2 x + 2 y \frac{d}{d x} [y])$

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

$10 (x^{2} + y^{2}) (2 x + 2 y y')$

Apply the distributive property.

$(10 x^{2} + 10 y^{2}) (2 x + 2 y y')$

Step 3

Differentiate the right side of the equation.

Differentiate.

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

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

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

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

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

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

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

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

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 $u$ as $y$ .

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

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

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

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

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

Multiply $2$ by $- 1$ .

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

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

$169 (2 x - 2 y y')$

Simplify.

Apply the distributive property.

$169 (2 x) + 169 (- 2 y y')$

Combine terms.

Multiply $2$ by $169$ .

$338 x + 169 (- 2 y y')$

Multiply $- 2$ by $169$ .

$338 x - 338 y y'$

Reorder terms.

$- 338 y y' + 338 x$

Step 4

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

$(10 x^{2} + 10 y^{2}) (2 x + 2 y y') = - 338 y y' + 338 x$

Step 5

Solve for $y'$ .

Simplify $(10 x^{2} + 10 y^{2}) (2 x + 2 y y')$ .

Rewrite.

$0 + 0 + (10 x^{2} + 10 y^{2}) (2 x + 2 y y') = - 338 y y' + 338 x$

Simplify by adding zeros.

$(10 x^{2} + 10 y^{2}) (2 x + 2 y y') = - 338 y y' + 338 x$

Expand $(10 x^{2} + 10 y^{2}) (2 x + 2 y y')$ using the FOIL Method.

Apply the distributive property.

$10 x^{2} (2 x + 2 y y') + 10 y^{2} (2 x + 2 y y') = - 338 y y' + 338 x$

Apply the distributive property.

$10 x^{2} (2 x) + 10 x^{2} (2 y y') + 10 y^{2} (2 x + 2 y y') = - 338 y y' + 338 x$

Apply the distributive property.

$10 x^{2} (2 x) + 10 x^{2} (2 y y') + 10 y^{2} (2 x) + 10 y^{2} (2 y y') = - 338 y y' + 338 x$

Simplify each term.

Rewrite using the commutative property of multiplication.

$10 \cdot 2 x^{2} x + 10 x^{2} (2 y y') + 10 y^{2} (2 x) + 10 y^{2} (2 y y') = - 338 y y' + 338 x$

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

Move $x$ .

$10 \cdot 2 (x \cdot x^{2}) + 10 x^{2} (2 y y') + 10 y^{2} (2 x) + 10 y^{2} (2 y y') = - 338 y y' + 338 x$

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

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

$10 \cdot 2 (x^{1} x^{2}) + 10 x^{2} (2 y y') + 10 y^{2} (2 x) + 10 y^{2} (2 y y') = - 338 y y' + 338 x$

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

$10 \cdot 2 x^{1 + 2} + 10 x^{2} (2 y y') + 10 y^{2} (2 x) + 10 y^{2} (2 y y') = - 338 y y' + 338 x$

Add $1$ and $2$ .

$10 \cdot 2 x^{3} + 10 x^{2} (2 y y') + 10 y^{2} (2 x) + 10 y^{2} (2 y y') = - 338 y y' + 338 x$

Multiply $10$ by $2$ .

$20 x^{3} + 10 x^{2} (2 y y') + 10 y^{2} (2 x) + 10 y^{2} (2 y y') = - 338 y y' + 338 x$

Multiply $2$ by $10$ .

$20 x^{3} + 20 x^{2} (y y') + 10 y^{2} (2 x) + 10 y^{2} (2 y y') = - 338 y y' + 338 x$

Rewrite using the commutative property of multiplication.

$20 x^{3} + 20 x^{2} y y' + 10 \cdot 2 y^{2} x + 10 y^{2} (2 y y') = - 338 y y' + 338 x$

Multiply $10$ by $2$ .

$20 x^{3} + 20 x^{2} y y' + 20 y^{2} x + 10 y^{2} (2 y y') = - 338 y y' + 338 x$

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

Move $y$ .

$20 x^{3} + 20 x^{2} y y' + 20 y^{2} x + 10 (y \cdot y^{2}) (2 y') = - 338 y y' + 338 x$

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

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

$20 x^{3} + 20 x^{2} y y' + 20 y^{2} x + 10 (y^{1} y^{2}) (2 y') = - 338 y y' + 338 x$

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

$20 x^{3} + 20 x^{2} y y' + 20 y^{2} x + 10 y^{1 + 2} (2 y') = - 338 y y' + 338 x$

Add $1$ and $2$ .

$20 x^{3} + 20 x^{2} y y' + 20 y^{2} x + 10 y^{3} (2 y') = - 338 y y' + 338 x$

Rewrite using the commutative property of multiplication.

$20 x^{3} + 20 x^{2} y y' + 20 y^{2} x + 10 \cdot 2 y^{3} y' = - 338 y y' + 338 x$

Multiply $10$ by $2$ .

$20 x^{3} + 20 x^{2} y y' + 20 y^{2} x + 20 y^{3} y' = - 338 y y' + 338 x$

Add $338 y y'$ to both sides of the equation.

$20 x^{3} + 20 x^{2} y y' + 20 y^{2} x + 20 y^{3} y' + 338 y y' = 338 x$

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

Subtract $20 x^{3}$ from both sides of the equation.

$20 x^{2} y y' + 20 y^{2} x + 20 y^{3} y' + 338 y y' = 338 x - 20 x^{3}$

Subtract $20 y^{2} x$ from both sides of the equation.

$20 x^{2} y y' + 20 y^{3} y' + 338 y y' = 338 x - 20 x^{3} - 20 y^{2} x$

Factor $2 y y'$ out of $20 x^{2} y y' + 20 y^{3} y' + 338 y y'$ .

Factor $2 y y'$ out of $20 x^{2} y y'$ .

$2 y y' (10 x^{2}) + 20 y^{3} y' + 338 y y' = 338 x - 20 x^{3} - 20 y^{2} x$

Factor $2 y y'$ out of $20 y^{3} y'$ .

$2 y y' (10 x^{2}) + 2 y y' (10 y^{2}) + 338 y y' = 338 x - 20 x^{3} - 20 y^{2} x$

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

$2 y y' (10 x^{2}) + 2 y y' (10 y^{2}) + 2 y y' \cdot 169 = 338 x - 20 x^{3} - 20 y^{2} x$

Factor $2 y y'$ out of $2 y y' (10 x^{2}) + 2 y y' (10 y^{2})$ .

$2 y y' (10 x^{2} + 10 y^{2}) + 2 y y' \cdot 169 = 338 x - 20 x^{3} - 20 y^{2} x$

Factor $2 y y'$ out of $2 y y' (10 x^{2} + 10 y^{2}) + 2 y y' \cdot 169$ .

$2 y y' (10 x^{2} + 10 y^{2} + 169) = 338 x - 20 x^{3} - 20 y^{2} x$

Divide each term in $2 y y' (10 x^{2} + 10 y^{2} + 169) = 338 x - 20 x^{3} - 20 y^{2} x$ by $2 y (10 x^{2} + 10 y^{2} + 169)$ and simplify.

Divide each term in $2 y y' (10 x^{2} + 10 y^{2} + 169) = 338 x - 20 x^{3} - 20 y^{2} x$ by $2 y (10 x^{2} + 10 y^{2} + 169)$ .

$\frac{2 y y' (10 x^{2} + 10 y^{2} + 169)}{2 y (10 x^{2} + 10 y^{2} + 169)} = \frac{338 x}{2 y (10 x^{2} + 10 y^{2} + 169)} + \frac{- 20 x^{3}}{2 y (10 x^{2} + 10 y^{2} + 169)} + \frac{- 20 y^{2} x}{2 y (10 x^{2} + 10 y^{2} + 169)}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

Rewrite the expression.

$\frac{y y' (10 x^{2} + 10 y^{2} + 169)}{y (10 x^{2} + 10 y^{2} + 169)} = \frac{338 x}{2 y (10 x^{2} + 10 y^{2} + 169)} + \frac{- 20 x^{3}}{2 y (10 x^{2} + 10 y^{2} + 169)} + \frac{- 20 y^{2} x}{2 y (10 x^{2} + 10 y^{2} + 169)}$

Cancel the common factor of $y$ .

Cancel the common factor.

Rewrite the expression.

$\frac{y' (10 x^{2} + 10 y^{2} + 169)}{10 x^{2} + 10 y^{2} + 169} = \frac{338 x}{2 y (10 x^{2} + 10 y^{2} + 169)} + \frac{- 20 x^{3}}{2 y (10 x^{2} + 10 y^{2} + 169)} + \frac{- 20 y^{2} x}{2 y (10 x^{2} + 10 y^{2} + 169)}$

Cancel the common factor of $10 x^{2} + 10 y^{2} + 169$ .

Cancel the common factor.

Divide $y'$ by $1$ .

$y' = \frac{338 x}{2 y (10 x^{2} + 10 y^{2} + 169)} + \frac{- 20 x^{3}}{2 y (10 x^{2} + 10 y^{2} + 169)} + \frac{- 20 y^{2} x}{2 y (10 x^{2} + 10 y^{2} + 169)}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $338$ and $2$ .

Factor $2$ out of $338 x$ .

$y' = \frac{2 (169 x)}{2 y (10 x^{2} + 10 y^{2} + 169)} + \frac{- 20 x^{3}}{2 y (10 x^{2} + 10 y^{2} + 169)} + \frac{- 20 y^{2} x}{2 y (10 x^{2} + 10 y^{2} + 169)}$

Cancel the common factors.

Factor $2$ out of $2 y (10 x^{2} + 10 y^{2} + 169)$ .

$y' = \frac{2 (169 x)}{2 (y (10 x^{2} + 10 y^{2} + 169))} + \frac{- 20 x^{3}}{2 y (10 x^{2} + 10 y^{2} + 169)} + \frac{- 20 y^{2} x}{2 y (10 x^{2} + 10 y^{2} + 169)}$

Cancel the common factor.

$y' = \frac{2 (169 x)}{2 (y (10 x^{2} + 10 y^{2} + 169))} + \frac{- 20 x^{3}}{2 y (10 x^{2} + 10 y^{2} + 169)} + \frac{- 20 y^{2} x}{2 y (10 x^{2} + 10 y^{2} + 169)}$

Rewrite the expression.

$y' = \frac{169 x}{y (10 x^{2} + 10 y^{2} + 169)} + \frac{- 20 x^{3}}{2 y (10 x^{2} + 10 y^{2} + 169)} + \frac{- 20 y^{2} x}{2 y (10 x^{2} + 10 y^{2} + 169)}$

Cancel the common factor of $- 20$ and $2$ .

Factor $2$ out of $- 20 x^{3}$ .

$y' = \frac{169 x}{y (10 x^{2} + 10 y^{2} + 169)} + \frac{2 (- 10 x^{3})}{2 y (10 x^{2} + 10 y^{2} + 169)} + \frac{- 20 y^{2} x}{2 y (10 x^{2} + 10 y^{2} + 169)}$

Cancel the common factors.

Factor $2$ out of $2 y (10 x^{2} + 10 y^{2} + 169)$ .

$y' = \frac{169 x}{y (10 x^{2} + 10 y^{2} + 169)} + \frac{2 (- 10 x^{3})}{2 (y (10 x^{2} + 10 y^{2} + 169))} + \frac{- 20 y^{2} x}{2 y (10 x^{2} + 10 y^{2} + 169)}$

Cancel the common factor.

$y' = \frac{169 x}{y (10 x^{2} + 10 y^{2} + 169)} + \frac{2 (- 10 x^{3})}{2 (y (10 x^{2} + 10 y^{2} + 169))} + \frac{- 20 y^{2} x}{2 y (10 x^{2} + 10 y^{2} + 169)}$

Rewrite the expression.

$y' = \frac{169 x}{y (10 x^{2} + 10 y^{2} + 169)} + \frac{- 10 x^{3}}{y (10 x^{2} + 10 y^{2} + 169)} + \frac{- 20 y^{2} x}{2 y (10 x^{2} + 10 y^{2} + 169)}$

Move the negative in front of the fraction.

$y' = \frac{169 x}{y (10 x^{2} + 10 y^{2} + 169)} - \frac{(10) x^{3}}{y (10 x^{2} + 10 y^{2} + 169)} + \frac{- 20 y^{2} x}{2 y (10 x^{2} + 10 y^{2} + 169)}$

Cancel the common factor of $- 20$ and $2$ .

Factor $2$ out of $- 20 y^{2} x$ .

$y' = \frac{169 x}{y (10 x^{2} + 10 y^{2} + 169)} - \frac{10 x^{3}}{y (10 x^{2} + 10 y^{2} + 169)} + \frac{2 (- 10 y^{2} x)}{2 y (10 x^{2} + 10 y^{2} + 169)}$

Cancel the common factors.

Factor $2$ out of $2 y (10 x^{2} + 10 y^{2} + 169)$ .

$y' = \frac{169 x}{y (10 x^{2} + 10 y^{2} + 169)} - \frac{10 x^{3}}{y (10 x^{2} + 10 y^{2} + 169)} + \frac{2 (- 10 y^{2} x)}{2 (y (10 x^{2} + 10 y^{2} + 169))}$

Cancel the common factor.

$y' = \frac{169 x}{y (10 x^{2} + 10 y^{2} + 169)} - \frac{10 x^{3}}{y (10 x^{2} + 10 y^{2} + 169)} + \frac{2 (- 10 y^{2} x)}{2 (y (10 x^{2} + 10 y^{2} + 169))}$

Rewrite the expression.

$y' = \frac{169 x}{y (10 x^{2} + 10 y^{2} + 169)} - \frac{10 x^{3}}{y (10 x^{2} + 10 y^{2} + 169)} + \frac{- 10 y^{2} x}{y (10 x^{2} + 10 y^{2} + 169)}$

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

Factor $y$ out of $- 10 y^{2} x$ .

$y' = \frac{169 x}{y (10 x^{2} + 10 y^{2} + 169)} - \frac{10 x^{3}}{y (10 x^{2} + 10 y^{2} + 169)} + \frac{y (- 10 y x)}{y (10 x^{2} + 10 y^{2} + 169)}$

Cancel the common factors.

Cancel the common factor.

$y' = \frac{169 x}{y (10 x^{2} + 10 y^{2} + 169)} - \frac{10 x^{3}}{y (10 x^{2} + 10 y^{2} + 169)} + \frac{y (- 10 y x)}{y (10 x^{2} + 10 y^{2} + 169)}$

Rewrite the expression.

$y' = \frac{169 x}{y (10 x^{2} + 10 y^{2} + 169)} - \frac{10 x^{3}}{y (10 x^{2} + 10 y^{2} + 169)} + \frac{- 10 y x}{10 x^{2} + 10 y^{2} + 169}$

Move the negative in front of the fraction.

$y' = \frac{169 x}{y (10 x^{2} + 10 y^{2} + 169)} - \frac{10 x^{3}}{y (10 x^{2} + 10 y^{2} + 169)} - \frac{10 y x}{10 x^{2} + 10 y^{2} + 169}$

To write $- \frac{10 y x}{10 x^{2} + 10 y^{2} + 169}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$y' = - \frac{10 x^{3}}{y (10 x^{2} + 10 y^{2} + 169)} + \frac{169 x}{y (10 x^{2} + 10 y^{2} + 169)} - \frac{10 y x}{10 x^{2} + 10 y^{2} + 169} \cdot \frac{y}{y}$

Write each expression with a common denominator of $y (10 x^{2} + 10 y^{2} + 169)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{10 y x}{10 x^{2} + 10 y^{2} + 169}$ by $\frac{y}{y}$ .

$y' = - \frac{10 x^{3}}{y (10 x^{2} + 10 y^{2} + 169)} + \frac{169 x}{y (10 x^{2} + 10 y^{2} + 169)} - \frac{10 y x y}{(10 x^{2} + 10 y^{2} + 169) y}$

Reorder the factors of $(10 x^{2} + 10 y^{2} + 169) y$ .

$y' = - \frac{10 x^{3}}{y (10 x^{2} + 10 y^{2} + 169)} + \frac{169 x}{y (10 x^{2} + 10 y^{2} + 169)} - \frac{10 y x y}{y (10 x^{2} + 10 y^{2} + 169)}$

Combine the numerators over the common denominator.

$y' = - \frac{10 x^{3}}{y (10 x^{2} + 10 y^{2} + 169)} + \frac{169 x - 10 y x y}{y (10 x^{2} + 10 y^{2} + 169)}$

Combine the numerators over the common denominator.

$y' = \frac{- 10 x^{3} + 169 x - 10 y x y}{y (10 x^{2} + 10 y^{2} + 169)}$

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

Move $y$ .

$y' = \frac{- 10 x^{3} + 169 x - 10 (y \cdot y) x}{y (10 x^{2} + 10 y^{2} + 169)}$

Multiply $y$ by $y$ .

$y' = \frac{- 10 x^{3} + 169 x - 10 y^{2} x}{y (10 x^{2} + 10 y^{2} + 169)}$

Factor $x$ out of $- 10 x^{3} + 169 x - 10 y^{2} x$ .

Factor $x$ out of $- 10 x^{3}$ .

$y' = \frac{x (- 10 x^{2}) + 169 x - 10 y^{2} x}{y (10 x^{2} + 10 y^{2} + 169)}$

Factor $x$ out of $169 x$ .

$y' = \frac{x (- 10 x^{2}) + x \cdot 169 - 10 y^{2} x}{y (10 x^{2} + 10 y^{2} + 169)}$

Factor $x$ out of $- 10 y^{2} x$ .

$y' = \frac{x (- 10 x^{2}) + x \cdot 169 + x (- 10 y^{2})}{y (10 x^{2} + 10 y^{2} + 169)}$

Factor $x$ out of $x (- 10 x^{2}) + x \cdot 169$ .

$y' = \frac{x (- 10 x^{2} + 169) + x (- 10 y^{2})}{y (10 x^{2} + 10 y^{2} + 169)}$

Factor $x$ out of $x (- 10 x^{2} + 169) + x (- 10 y^{2})$ .

$y' = \frac{x (- 10 x^{2} + 169 - 10 y^{2})}{y (10 x^{2} + 10 y^{2} + 169)}$

Factor $- 1$ out of $- 10 x^{2}$ .

$y' = \frac{x (- (10 x^{2}) + 169 - 10 y^{2})}{y (10 x^{2} + 10 y^{2} + 169)}$

Rewrite $169$ as $- 1 (- 169)$ .

$y' = \frac{x (- (10 x^{2}) - 1 (- 169) - 10 y^{2})}{y (10 x^{2} + 10 y^{2} + 169)}$

Factor $- 1$ out of $- (10 x^{2}) - 1 (- 169)$ .

$y' = \frac{x (- (10 x^{2} - 169) - 10 y^{2})}{y (10 x^{2} + 10 y^{2} + 169)}$

Factor $- 1$ out of $- 10 y^{2}$ .

$y' = \frac{x (- (10 x^{2} - 169) - (10 y^{2}))}{y (10 x^{2} + 10 y^{2} + 169)}$

Factor $- 1$ out of $- (10 x^{2} - 169) - (10 y^{2})$ .

$y' = \frac{x (- (10 x^{2} - 169 + 10 y^{2}))}{y (10 x^{2} + 10 y^{2} + 169)}$

Simplify the expression.

Rewrite $- (10 x^{2} - 169 + 10 y^{2})$ as $- 1 (10 x^{2} - 169 + 10 y^{2})$ .

$y' = \frac{x (- 1 (10 x^{2} - 169 + 10 y^{2}))}{y (10 x^{2} + 10 y^{2} + 169)}$

Move the negative in front of the fraction.

$y' = - \frac{x (10 x^{2} - 169 + 10 y^{2})}{y (10 x^{2} + 10 y^{2} + 169)}$

Step 6

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

$\frac{d y}{d x} = - \frac{x (10 x^{2} - 169 + 10 y^{2})}{y (10 x^{2} + 10 y^{2} + 169)}$

Find dy/dx 5(x^2+y^2)^2=169(x^2-y^2)

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