18. Chain Rule

Dated: 30-10-2024

Chain Rule is used for differentiation of composite functions.
Say, we have a function¹ y such that

\[y = (f o g)(x) = f\left(g(x)\right)\]

Then assume, \(g(x) = u\) then \((fog)(x)\) takes the form \(f(u)\).
Then we have:

\[\frac{dy}{du} = f^{'}(u)\]

\[\frac{du}{dx} = g^{'}(x)\]

Hence, the rule itself is defined as:

\[\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = f^{'}(u) \cdot g^{'}(x)\]

References