Math / Calculus

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If $-2=x y+5 y+3 x^{3}$ then find $\frac{d y}{d x}$ in terms of $x$ and $y$ .
Answer Attempt 1 out of 2

Studdy Solution

STEP 1

1. The equation $-2 = xy + 5y + 3x^3$ is implicitly defined.
2. We need to find the derivative $\frac{dy}{dx}$ using implicit differentiation.

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 $-2 = xy + 5y + 3x^3$ with respect to $x$ .
The left side is a constant: $\frac{d}{dx}(-2) = 0$
Differentiate the right side term by term: $\frac{d}{dx}(xy) = x \frac{dy}{dx} + y$ $\frac{d}{dx}(5y) = 5 \frac{dy}{dx}$ $\frac{d}{dx}(3x^3) = 9x^2$
Combine these results: $0 = x \frac{dy}{dx} + y + 5 \frac{dy}{dx} + 9x^2$

STEP 4

Rearrange the differentiated equation to solve for $\frac{dy}{dx}$ .
Combine the $\frac{dy}{dx}$ terms: $0 = (x + 5) \frac{dy}{dx} + y + 9x^2$
Isolate $\frac{dy}{dx}$ : $(x + 5) \frac{dy}{dx} = -y - 9x^2$
Divide both sides by $x + 5$ : $\frac{dy}{dx} = \frac{-y - 9x^2}{x + 5}$
The derivative $\frac{dy}{dx}$ in terms of $x$ and $y$ is:
$\frac{dy}{dx} = \frac{-y - 9x^2}{x + 5}$

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