Find dy/dx 8x^2+y^2=5

$8 x^{2} + y^{2} = 5$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (8 x^{2} + y^{2}) = \frac{d}{d x} (5)$

Step 2

Differentiate the left side of the equation.

By the Sum Rule, the derivative of $8 x^{2} + y^{2}$ with respect to $x$ is $\frac{d}{d x} [8 x^{2}] + \frac{d}{d x} [y^{2}]$ .

$\frac{d}{d x} [8 x^{2}] + \frac{d}{d x} [y^{2}]$

Evaluate $\frac{d}{d x} [8 x^{2}]$ .

Since $8$ is constant with respect to $x$ , the derivative of $8 x^{2}$ with respect to $x$ is $8 \frac{d}{d x} [x^{2}]$ .

$8 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [y^{2}]$

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

$8 (2 x) + \frac{d}{d x} [y^{2}]$

Multiply $2$ by $8$ .

$16 x + \frac{d}{d x} [y^{2}]$

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

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 $y$ .

$16 x + \frac{d}{d u} [u^{2}] \frac{d}{d x} [y]$

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

$16 x + 2 u \frac{d}{d x} [y]$

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

$16 x + 2 y \frac{d}{d x} [y]$

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

$16 x + 2 y y'$

Reorder terms.

$2 y y' + 16 x$

Step 3

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$0$

Step 4

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

$2 y y' + 16 x = 0$

Step 5

Solve for $y'$ .

Subtract $16 x$ from both sides of the equation.

$2 y y' = - 16 x$

Divide each term in $2 y y' = - 16 x$ by $2 y$ and simplify.

Divide each term in $2 y y' = - 16 x$ by $2 y$ .

$\frac{2 y y'}{2 y} = \frac{- 16 x}{2 y}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 y y'}{2 y} = \frac{- 16 x}{2 y}$

Rewrite the expression.

$\frac{y y'}{y} = \frac{- 16 x}{2 y}$

Cancel the common factor of $y$ .

Cancel the common factor.

$\frac{y y'}{y} = \frac{- 16 x}{2 y}$

Divide $y'$ by $1$ .

$y' = \frac{- 16 x}{2 y}$

Simplify the right side.

Cancel the common factor of $- 16$ and $2$ .

Factor $2$ out of $- 16 x$ .

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

Cancel the common factors.

Factor $2$ out of $2 y$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Step 6

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

$\frac{d y}{d x} = - \frac{8 x}{y}$

Find dy/dx 8x^2+y^2=5

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