So far, we have developed algorithms for differentiation for a growing
list of functions-powers, roots, exponentials, and trig
functions-and we have developed methods for differentiating more
complicated functions that are constructed from these functions by
algebraic operations-addition, subtraction, multiplication, and
division. Here we would like to add to our catalogue of techniques by
considering functions that are compositions of other functions.
If z = g(x) and y = f(z), we can define a new function, y=h(x) by
the function h is called the composition of f and g. If both f and g are differentiable, the composition h is also differentiable. The derivative of h is given by
In words, the derivative of a composition is the product of the derivative
of the outside function evaluated at the inside function and the derivative
of the inside function.
For example, if 
, h is the composition of
and 
. Since 
and
g'(x) = 2x,
The following exercises will provide some practice with the chain rule.