The Chain Rule

The chain rule is learned in the first semester of calculus and tends to be a rule that many students find confusing. Like many things in math, it's much simpler in reality than it first appears. The chain rule is used for taking the derivative of composite functions. As a reminder composite functions are when you plug one function into another. So for example let's say you wanted to take the derivative of y=sin(x^2). This is a composite function. We call the outside function f(w) = sin(w) and the inside function g(x) = x^2. Note that f(g(x)) = sin(x^2) which is our original function. I find the easiest way to interpret the chain rule is to write the instructions into a sentence. That sentence is "Take the derivative of the outside function, leave the inside function plugged into the derivative of the outside function. Then multiply the result by the derivative of the inside function." Let's use this on our example y = sin(x^2). The derivative of the outside function is cos(w). Then we leave the inside function plugged into the derivative we just took to get cos(x^2). We then multiply the result by the derivative of the inside function to get cos(x^2)*(2x). And there you have it, the derivative of sin(x^2) is 2x*cos(x^2).