Implicit differentiation worksheet (with detailed solutions) - Free Printable

- Problem 14: $y^2 - x\sqrt{x^2 - 3} = 0$
- Differentiate both sides with respect to $x$:
$$
\frac{d}{dx}(y^2) - \frac{d}{dx}\left(x\sqrt{x^2 - 3}\right) = 0
$$
- Apply chain rule and product rule:
$$
2y \frac{dy}{dx} - \left[\sqrt{x^2 - 3} + x \cdot \frac{1}{2}(x^2 - 3)^{-1/2} \cdot 2x\right] = 0
$$
- Simplify:
$$
2y \frac{dy}{dx} - \left[\sqrt{x^2 - 3} + \frac{x^2}{\sqrt{x^2 - 3}}\right] = 0
$$
- Combine terms:
$$
2y \frac{dy}{dx} = \frac{(x^2 - 3) + x^2}{\sqrt{x^2 - 3}} = \frac{2x^2 - 3}{\sqrt{x^2 - 3}}
$$
- Solve for $\frac{dy}{dx}$:
$$
\frac{dy}{dx} = \frac{2x^2 - 3}{2y\sqrt{x^2 - 3}}
$$

- Problem 15: $x^3 + y^3 - 9xy = 0$
- Differentiate both sides with respect to $x$:
$$
\frac{d}{dx}(x^3) + \frac{d}{dx}(y^3) - \frac{d}{dx}(9xy) = 0
$$
- Apply chain rule and product rule:
$$
3x^2 + 3y^2 \frac{dy}{dx} - 9\left(y + x \frac{dy}{dx}\right) = 0
$$
- Expand and group terms:
$$
3x^2 + 3y^2 \frac{dy}{dx} - 9y - 9x \frac{dy}{dx} = 0
$$
- Factor $\frac{dy}{dx}$:
$$
(3y^2 - 9x)\frac{dy}{dx} = 9y - 3x^2
$$
- Solve for $\frac{dy}{dx}$:
$$
\frac{dy}{dx} = \frac{9y - 3x^2}{3y^2 - 9x} = \frac{3y - x^2}{y^2 - 3x}
$$

- Problem 16: $x^2 + 3y^2 = 10$
- Differentiate both sides with respect to $x$:
$$
\frac{d}{dx}(x^2) + \frac{d}{dx}(3y^2) = \frac{d}{dx}(10)
$$
- Apply chain rule:
$$
2x + 6y \frac{dy}{dx} = 0
$$
- Solve for $\frac{dy}{dx}$:
$$
6y \frac{dy}{dx} = -2x \quad \Rightarrow \quad \frac{dy}{dx} = -\frac{x}{3y}
$$

- Problem 17: $e^x y = e^y$
- Differentiate both sides with respect to $x$:
$$
\frac{d}{dx}(e^x y) = \frac{d}{dx}(e^y)
$$
- Apply product rule and chain rule:
$$
e^x y + e^x \frac{dy}{dx} = e^y \frac{dy}{dx}
$$
- Rearrange terms:
$$
e^x y = e^y \frac{dy}{dx} - e^x \frac{dy}{dx} = \frac{dy}{dx}(e^y - e^x)
$$
- Solve for $\frac{dy}{dx}$:
$$
\frac{dy}{dx} = \frac{e^x y}{e^y - e^x}
$$