Math / Calculus

QuestionIf $-x y=3 y^{3}+2 x^{2}$ then find $\frac{d y}{d x}$ in terms of $x$ and $y$ .

Studdy Solution

STEP 1

1. The equation $-x y = 3 y^3 + 2 x^2$ is implicitly defined, and we need to find $\frac{dy}{dx}$ .
2. We will use implicit differentiation to find the derivative.

STEP 2

1. Differentiate both sides of the equation with respect to $x$ .
2. Solve for $\frac{dy}{dx}$ .

STEP 3

Differentiate both sides of the equation with respect to $x$ . We will apply the product rule and chain rule as necessary.
The left side of the equation is $-xy$ . Using the product rule, the derivative is:
$\frac{d}{dx}(-xy) = -\left(x \frac{dy}{dx} + y\right)$
The right side of the equation is $3y^3 + 2x^2$ . Differentiate each term separately:
$\frac{d}{dx}(3y^3) = 9y^2 \frac{dy}{dx}$
$\frac{d}{dx}(2x^2) = 4x$
Putting it all together, we have:
$-x \frac{dy}{dx} - y = 9y^2 \frac{dy}{dx} + 4x$

STEP 4

Solve for $\frac{dy}{dx}$ . First, collect all terms involving $\frac{dy}{dx}$ on one side of the equation:
$-x \frac{dy}{dx} - 9y^2 \frac{dy}{dx} = 4x + y$
Factor out $\frac{dy}{dx}$ :
$\frac{dy}{dx}(-x - 9y^2) = 4x + y$
Solve for $\frac{dy}{dx}$ :
$\frac{dy}{dx} = \frac{4x + y}{-x - 9y^2}$
The derivative $\frac{dy}{dx}$ in terms of $x$ and $y$ is:
$\boxed{\frac{4x + y}{-x - 9y^2}}$

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