Question

Explain the chain rule in differential calculus with some examples to make it clear :) ​

Answer 1

Answer:

The chain rule states that the derivative of f(g(x)) is f'(g(x))⋅g'(x). In other words, it helps us differentiate *composite functions*. For example, sin(x²) is a composite function because it can be constructed as f(g(x)) for f(x)=sin(x) and g(x)=x².

Answer 2

Hey Mylo!

Let's try to figure out how the chain rule exactly works.

Consider a function y = log(e^sin(x)). This is a composition of three functions that are: the logarithmic function, the exponential function and the sine function.

Let's try to differentiate it.

To make this composition of functions look quite simpler, let e^sin(x) = u and sin(x) = v

Thus, the function becomes:

y = log(u) .......................................(a)

Now, differentiate (a) wrt u,

=> dy/du = 1/u = 1/e^sin(x) .........(i)

also,

u = e^sin(x) = e^v .......................(b)

differentiate (b) wrt v,

du/dv = e^v = e^sin(x) .................(ii)

and, v = sin(x) ................................(c)

differentiate (c) wrt x,

dv/dx = cos(x) ...............................(iii)

Multiply (i), (ii) and (iii),

dy/du × du/dv × dv/dx = 1/e^sin(x) ×e^sin(x) × cos(x)

=> dy/dx = cos(x)

That is, differentiation of the considered function is cos(x).

So, this was how the chain rule works.

Still got a doubt? Check out your inbox! Ask me there. I'll seek to help ya in anyway I can! Cheers!