Chain Rule

Let y=f(u)   be a differentiable function of u  and u=g(x)  be a differentiable function of x . Then y=f(g(x))  is a diffferentiable function of x . And,

dy dx = f (g(x)) g (x)

This is called the chain rule. To see that it is indeed true, we try to find the derivative of y  with respect to x  using the definition of the derivative.

dy dx = lim Δx0 Δy Δx

= lim Δx0 Δy Δu Δu Δx

=( lim Δx0 Δy Δu )( lim Δx0 Δu Δx )

=( lim Δu0 Δy Δu )( lim Δx0 Δu Δx )

This is because since u   is a differentiable function of x  , we have Δu0   whenever Δx0

= dy du du dx

= f (u) g (x)

= f (g(x)) g (x)

Note that we have

dy dx = dy du du dx

This is another way to write the chain rule. And it helps to write down the chain rule when we are dealing with a chain of function as below.

Suppose y   is a differentiable function of u   given by y=f(u)  , u   in turn is a differentiable function of x   given by u=g(x)  , and x   in turn is a differentiable function of t   given by x=h(t)  , then y   is a differentiable function of t   and

dy dt = dy du du dx du dt

This is much easier to write than the more complex looking form below

dy dt =[f'(g(h(t))][ g (h(t))][ h (t)]

Note that both are same written in different form as you can see from below.

dy du = f (u)= f (g(x))= f (g(h(t)))

du dx = g (x)= g (h(t))

dx dt = h (t)