1. $ x^2 + y^2 = 9 $
- Differentiate both sides with respect to $ x $:
$$
\frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) = \frac{d}{dx}(9)
$$
$$
2x + 2y \frac{dy}{dx} = 0
$$
- Solve for $ \frac{dy}{dx} $:
$$
2y \frac{dy}{dx} = -2x
$$
$$
\frac{dy}{dx} = -\frac{x}{y}
$$
3. $ x^{1/2} + y^{1/2} = 16 $
- Differentiate both sides with respect to $ x $:
$$
\frac{d}{dx}(x^{1/2}) + \frac{d}{dx}(y^{1/2}) = \frac{d}{dx}(16)
$$
$$
\frac{1}{2}x^{-1/2} + \frac{1}{2}y^{-1/2} \frac{dy}{dx} = 0
$$
- Solve for $ \frac{dy}{dx} $:
$$
\frac{1}{2}y^{-1/2} \frac{dy}{dx} = -\frac{1}{2}x^{-1/2}
$$
$$
\frac{dy}{dx} = -\frac{x^{-1/2}}{y^{-1/2}} = -\sqrt{\frac{y}{x}}
$$
Parent Tip: Review the logic above to help your child master the concept of implicit differentiation worksheet.
