Implicit differentiation is the [[Derivative|differentiation]] of both sides of an equation with respect to a variable $x$, and then solving the resulting equation for $\frac{dy}{dx}$ that emerged from the application of the [[Math/Calculus/Derivative/Chain Rule|chain rule]]. It is especially useful for differentiating implicit curves, and functions of which their [[Inverse|inverses]] are known. $ \begin{align*} y &= f(x) \\ f^{-1}(y) &= x\\ \frac{d}{dx}(f^{-1}(y)) &= \frac{d}{dx}x \\ \frac{dy}{dx}\frac{d}{dy}(f^{-1}(y)) &= 1 \\ \frac{dy}{dx} &= \frac{1}{\frac{d}{dy}(f^{-1}(y))} \end{align*} $