Question

(b) If $2x^2 - 5xy + y^3 = 76$, then find $\frac{dy}{dx}$.

Hint:

When differentiating implicit equations, remember to apply the product rule and chain rule where necessary.

Answer 1

The implicit differentiation of the given equation with respect to $x$ is performed. The equation is: \[ 2x^2 - 5xy + y^3 = 76 \] Differentiating each term yields: \[ \frac{d}{dx}(2x^2) - \frac{d}{dx}(5xy) + \frac{d}{dx}(y^3) = \frac{d}{dx}(76) \] The derivatives are calculated as: \[ 4x - 5\left( \frac{d}{dx}(x) y + x \frac{d}{dx}(y) \right) + 3y^2 \frac{dy}{dx} = 0 \] After simplification: \[ 4x - 5(y + x \frac{dy}{dx}) + 3y^2 \frac{dy}{dx} = 0 \] Solving for $\frac{dy}{dx}$: \[ 4x - 5y - 5x \frac{dy}{dx} + 3y^2 \frac{dy}{dx} = 0 \] \[ - 5x \frac{dy}{dx} + 3y^2 \frac{dy}{dx} = 5y - 4x \] Factoring out $\frac{dy}{dx}$: \[ \frac{dy}{dx} \left( -5x + 3y^2 \right) = 5y - 4x \] The final result is: \[ \frac{dy}{dx} = \frac{5y - 4x}{3y^2 - 5x} \]