A common misconception leads to an incorrect calculation of the derivative of a composition of functions. This demo contrasts the resulting error with the correct answer, given by the chain rule.
The first window graphs two functions: f(x) (in gray) and its derivative, f '(x) (in green).
In the second window, we see three curves, a graph of the composition f(g(x)) (in gray), an incorrect calculation of the derivative of this composed function with respect to x (in green), and a correct calculation (in red).
The incorrect calculation, common among those unfamiliar with the chain rule, is given by f '(g(x)). (For example, one might be inclined to think that the derivative with respect to x of sin(2x) is cos(2x)).
One useful way to think about the chain rule is to consider the rate at which we are "moving through" the values of the function f(x). For example, though sin(x) and sin(2x) both express rates of change as one moves through the values of the sine function, the function sin(2x) goes through those values twice as fast as does the function sin(x), which makes the graph twice as steep, and requires us to correct our mistake (cos(2x)) by multiplying by 2. The "correction factor" of g' (x) tells us how fast we are moving through the values of the function f(x).
The correct graph of the derivative of f(g(x)) with respect to x is shown in red.