Find dx/dy 5x^2+xy+5y^2=11

$5 x^{2} + x y + 5 y^{2} = 11$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (5 x^{2} + x y + 5 y^{2}) = \frac{d}{d y} (11)$

Step 2

Differentiate the left side of the equation.

By the Sum Rule, the derivative of $5 x^{2} + x y + 5 y^{2}$ with respect to $y$ is $\frac{d}{d y} [5 x^{2}] + \frac{d}{d y} [x y] + \frac{d}{d y} [5 y^{2}]$ .

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

Evaluate $\frac{d}{d y} [5 x^{2}]$ .

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

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

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

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

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

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

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

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

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

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

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

Multiply $2$ by $5$ .

$10 x x' + \frac{d}{d y} [x y] + \frac{d}{d y} [5 y^{2}]$

Evaluate $\frac{d}{d y} [x y]$ .

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = x$ and $g (y) = y$ .

$10 x x' + x \frac{d}{d y} [y] + y \frac{d}{d y} [x] + \frac{d}{d y} [5 y^{2}]$

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

$10 x x' + x \cdot 1 + y \frac{d}{d y} [x] + \frac{d}{d y} [5 y^{2}]$

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

$10 x x' + x \cdot 1 + y x' + \frac{d}{d y} [5 y^{2}]$

Multiply $x$ by $1$ .

$10 x x' + x + y x' + \frac{d}{d y} [5 y^{2}]$

Evaluate $\frac{d}{d y} [5 y^{2}]$ .

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

$10 x x' + x + y x' + 5 \frac{d}{d y} [y^{2}]$

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

$10 x x' + x + y x' + 5 (2 y)$

Multiply $2$ by $5$ .

$10 x x' + x + y x' + 10 y$

Reorder terms.

$10 x x' + y x' + x + 10 y$

Step 3

Since $11$ is constant with respect to $y$ , the derivative of $11$ with respect to $y$ is $0$ .

$0$

Step 4

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

$10 x x' + y x' + x + 10 y = 0$

Step 5

Solve for $x'$ .

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

Subtract $x$ from both sides of the equation.

$10 x x' + y x' + 10 y = - x$

Subtract $10 y$ from both sides of the equation.

$10 x x' + y x' = - x - 10 y$

Factor $x'$ out of $10 x x' + y x'$ .

Factor $x'$ out of $10 x x'$ .

$x' (10 x) + y (x') = - x - 10 y$

Factor $x'$ out of $y x'$ .

$x' (10 x) + x' (y) = - x - 10 y$

Factor $x'$ out of $x' (10 x) + x' (y)$ .

$x' (10 x + y) = - x - 10 y$

Divide each term in $x' (10 x + y) = - x - 10 y$ by $10 x + y$ and simplify.

Divide each term in $x' (10 x + y) = - x - 10 y$ by $10 x + y$ .

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

Simplify the left side.

Cancel the common factor of $10 x + y$ .

Cancel the common factor.

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

Divide $x'$ by $1$ .

$x' = \frac{- x}{10 x + y} + \frac{- 10 y}{10 x + y}$

Simplify the right side.

Combine the numerators over the common denominator.

$x' = \frac{- x - 10 y}{10 x + y}$

Factor $- 1$ out of $- x$ .

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

Factor $- 1$ out of $- 10 y$ .

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

Factor $- 1$ out of $- (x) - (10 y)$ .

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

Simplify the expression.

Rewrite $- (x + 10 y)$ as $- 1 (x + 10 y)$ .

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

Move the negative in front of the fraction.

$x' = - \frac{x + 10 y}{10 x + y}$

Step 6

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

$\frac{d x}{d y} = - \frac{x + 10 y}{10 x + y}$

Find dx/dy 5x^2+xy+5y^2=11

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