Question

Find \( \frac{dy}{dx} \) if \( x^3 + y^3 + x^2 = a^b \), where \( a \) and \( b \) are constants.

Hint:

For implicit differentiation, treat \( y \) as a function of \( x \) and apply the chain rule when differentiating terms involving \( y \).

Answer 1

Given the equation \( x^3 + y^3 + x^2 = a^b \), we will find \( \frac{dy}{dx} \) by implicit differentiation with respect to \( x \). Differentiating each term: \[ \frac{d}{dx}(x^3) + \frac{d}{dx}(y^3) + \frac{d}{dx}(x^2) = \frac{d}{dx}(a^b) \] Since \( a^b \) is a constant, its derivative is 0. Thus, \[ 3x^2 + 3y^2 \frac{dy}{dx} + 2x = 0 \] Isolating \( \frac{dy}{dx} \): \[ 3y^2 \frac{dy}{dx} = -3x^2 - 2x \] \[ \frac{dy}{dx} = \frac{-3x^2 - 2x}{3y^2} \]