Find dx/dy at (12,3) x^2+4y^2=36 , (12,3)

$x^{2} + 4 y^{2} = 36$ , $(12, 3)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (x^{2} + 4 y^{2}) = \frac{d}{d y} (36)$

Step 2

Differentiate the left side of the equation.

By the Sum Rule, the derivative of $x^{2} + 4 y^{2}$ with respect to $y$ is $\frac{d}{d y} [x^{2}] + \frac{d}{d y} [4 y^{2}]$ .

$\frac{d}{d y} [x^{2}] + \frac{d}{d y} [4 y^{2}]$

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

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 $x$ .

$\frac{d}{d u} [u^{2}] \frac{d}{d y} [x] + \frac{d}{d y} [4 y^{2}]$

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

$2 u \frac{d}{d y} [x] + \frac{d}{d y} [4 y^{2}]$

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

$2 x \frac{d}{d y} [x] + \frac{d}{d y} [4 y^{2}]$

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

$2 x x' + \frac{d}{d y} [4 y^{2}]$

Evaluate $\frac{d}{d y} [4 y^{2}]$ .

Since $4$ is constant with respect to $y$ , the derivative of $4 y^{2}$ with respect to $y$ is $4 \frac{d}{d y} [y^{2}]$ .

$2 x x' + 4 \frac{d}{d y} [y^{2}]$

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

$2 x x' + 4 (2 y)$

Multiply $2$ by $4$ .

$2 x x' + 8 y$

Step 3

Since $36$ is constant with respect to $y$ , the derivative of $36$ with respect to $y$ is $0$ .

$0$

Step 4

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

$2 x x' + 8 y = 0$

Step 5

Solve for $x'$ .

Subtract $8 y$ from both sides of the equation.

$2 x x' = - 8 y$

Divide each term in $2 x x' = - 8 y$ by $2 x$ and simplify.

Divide each term in $2 x x' = - 8 y$ by $2 x$ .

$\frac{2 x x'}{2 x} = \frac{- 8 y}{2 x}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x x'}{2 x} = \frac{- 8 y}{2 x}$

Rewrite the expression.

$\frac{x x'}{x} = \frac{- 8 y}{2 x}$

Cancel the common factor of $x$ .

Cancel the common factor.

$\frac{x x'}{x} = \frac{- 8 y}{2 x}$

Divide $x'$ by $1$ .

$x' = \frac{- 8 y}{2 x}$

Simplify the right side.

Cancel the common factor of $- 8$ and $2$ .

Factor $2$ out of $- 8 y$ .

$x' = \frac{2 (- 4 y)}{2 x}$

Cancel the common factors.

Factor $2$ out of $2 x$ .

$x' = \frac{2 (- 4 y)}{2 (x)}$

Cancel the common factor.

$x' = \frac{2 (- 4 y)}{2 x}$

Rewrite the expression.

$x' = \frac{- 4 y}{x}$

Move the negative in front of the fraction.

$x' = - \frac{4 y}{x}$

Step 6

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

$\frac{d x}{d y} = - \frac{4 y}{x}$

Step 7

Replace $x$ with $12$ and $y$ with $3$ in the expression.

$\frac{d x}{d y} = - \frac{4 (3)}{12}$

Step 8

Simplify the result.

Cancel the common factors.

Factor $4$ out of $12$ .

$- \frac{4 \cdot 3}{4 \cdot 3}$

Cancel the common factor.

$- \frac{4 \cdot 3}{4 \cdot 3}$

Rewrite the expression.

$- \frac{3}{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$- \frac{3}{3}$

Rewrite the expression.

$- 1 \cdot 1$

Multiply $- 1$ by $1$ .

$- 1$

Find dx/dy at (12,3) x^2+4y^2=36 , (12,3)

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