19. Implicit Differentiation

Dated: 30-10-2024

Imagine we have a function¹ defined as:

\[yx = 1\]

We can differentiate it by solving for y first.

\[y = \frac{1}{x}\]

\[\frac{d}{dx} y = \frac{d}{dx} \left(\frac{1}{x}\right)\]

\[\frac{dy}{dx} = -\frac{1}{x^2}\]

Alternatively, we can also do it by without solving for y.

\[yx = 1\]

Applying chain rule.²

\[x\frac{d}{dx}(y)+y\frac{d}{dx}(x)=0\]

\[x\frac{dy}{dx}+y(1)=0\]

\[x\frac{dy}{dx}=-y\]

\[\frac{dy}{dx}=-\frac{y}{x}\]

Then we find the value of y and substitute it.
From the original equation, we get:

\[y = \frac{1}{x}\]

Place it in the previous equation and we get.

\[\frac{dy}{dx} = -\frac{1}{x^2}\]

This method of differentiation without finding value of y is called implicit differentiation.

References