Implicit Differentiation

Why

Used when getting an explicit expression for $y$ in terms of $x$ is difficult

How

1. Differentiate both sides of the equation W.R.T. $x$
   • $\frac{d}{dx}g(y)=g^{\prime }(y)\frac{dy}{dx}$
2. Solve the resulting equation for $\frac{dy}{dx}$

Example

Find $\frac{dy}{dx}$ for $x^3+y^3-9xy=0$ Differentiating both sides W.R.T. $x$, $3x^2+3y^2\frac{dy}{dx}-9y-9x\frac{dy}{dx}=0$ Solving for $\frac{dy}{dx}$, $(3y^2-9x)\frac{dy}{dx}=-3x^2+9y$ $\Rightarrow \frac{dy}{dx}=-\frac{3x^2-9y}{3y^2-9x}$