Note that it’s not scientific terms but just informal way to call functions for better understanding. Therefore we’ll call function g an outside function and function f an inside function.

Here we have two functions g and f and function f, so to speak, is enclosed into function g. To obtain value of function h at some point x we should first apply function f then we apply g to the obtained result. In real analysis, it’s proved that if function f has derivative at the point x_0 and function g has derivative at the point f(x_0) then composite function h(x)=g(f(x)) also has a derivative at the point x_0.Ĭhain rule states that the derivative of composite function h(x) is found as follows:

For example, consider the following function:Ĭomposite function h(x) is a composition of functions f and g.

Composition of functionsįirst of all, what is composition of functions? Actually it means function of another function. This rule is needed when we want to find derivative of non-common function which is in fact composition of two or more common functions. Let’s discuss chain rule for derivatives or so called outside-inside rule. But what if given expression is more complicated? That’s what this section is about. If you’re lucky and the function is among common functions, you can write down the required derivative right away using table of derivatives if necessary. Take the derivative of the following functions, each of which involves the chain rule.If you’re asked to find a derivative of some function, sometimes it’s not a trivial thing to do.

The chain rule homework