The product rule tells us how to find the derivative of a product of functions like f(x) • g(x). The composition or “chain” rule tells us how to find the derivative of a composition of functions like f(g(x)). Composition of functions is about substitution – you substitute a value for x into the formula for g, then you substitute the result into the formula for f.
One way to think about composition of functions is to use new variable names. It’s good practice to introduce new variables when they’re convenient, and this is one place where it’s very convenient. So, how do we find the derivative of a composition of functions? We’re trying to find the slope of a tangent line; to do this we take a limit of slopes Δy of Δt (also can be called as (Δy)/(Δt)) secant lines. Here y is a function of x, x is a function of t, and we want to know how y changes with respect to the original variable t. Here again using that intermediate variable x is a big help:
(Δy)/(Δt) = (Δy)/(Δx) times (x)/(Δt)
because when we perform the multiplication, the small change Δx cancels.
The derivative of y with respect to t is “the limit as Δt approaches 0 of (Δy)/(Δt).”
Here is another way of writing the chain rule: d/dx (f◦g)(x) = d/dx (f(g(x))) = f’(g(x)) g’(x)