JEE Main & Advanced
Suppose we had to differentiate f(x) = (3x2 + 4x - 5)^2. We may be inclined to multiply out this polynomial, and then differentiate it out term by term. However, as lazy mathematicians, we are always looking for tricks to make life easier. Here, we can use the chain rule.
The chain rule helps us find the derivative of functions compounded inside of one of another. In our example, f(x) = (3x2 + 4x - 5)^2, we have our polynomial, 3x2 + 4x - 5, inside our outer function, (something)^2. Now that we've identified this as a compound function, we can use the chain rule to find that
f'(x) = 2(3x2 + 4x - 5)(6x + 4).
Chain rule: [f(g(x))]' = f'(g(x)) * g'(x)
f(x) = (3x2 + 4x - 5)^2
f'(x) = 2(3x2 + 4x - 5) * (6x + 4)
In conclusion, we use the chain rule to make differentiating easier (and, as we'll see, in some cases it allows us to find derivatives where we couldn't before). ANYTIME that you see a function nested inside of another function, think to use the chain rule. It is incredibly useful. Imagine, for example, if we had tried to multiply out f(x) = (3x2 + 4x - 5)^2 and then differentiated it. Slightly manageable, right? Now, imagine if we tried to multiply out f(x) = (3x2 + 4x - 5)^5. Not worth the time, but with the chain rule it becomes incredibly easy.