In this section we derive the formula for the derivative of the composition of two functions.

The Chain Rule: If g'(x) and f'(g(x)) both exist, and F=fog is the composition F(x) = f(g(x)), then F'(x) exists, and is given by

F'(x) = f'(g(x))g'(x)

or alternately, letting y=f(u) and u=g(x),

dy/dx = dy/du du/dx

The key idea is to write the quotient

f(g(x+h))-f(g(x))/h

as

[f(g(x+h))-f(g(x))/(g(x+h)-g(x))].[(g(x+h)-g(x))/h]

This naturally generates two terms f'(g(x)) and g'(x).

Note: is sometimes easier to actually compute the composition, f(g(x)) and then differentiate the result.

Special Case: The Power Rule - d/dx (u ⁿ ) = n u ^n-1 du/dx

d/dx [g(x)] ⁿ = n [g(x)] ^n-1 g'(x)

Last modified Wed Oct 2 21:47:54 CDT 1996