Math Help

Derivatives – Using the Chain Rule (Very Important)

In a Nut Shell: For more complicated functions, f(x), it is often easier to take a

“chain” of derivatives of simpler functions and multiply them together to get the

derivative of the original complicated function, f(x). In so doing, it is best to identify

the dependent variable, y, the independent variable, x, and perhaps, introduce an

intermediate variable, u.

Consider the more complicated function y(x) = y( u(x) )

which reads as y is a function of u which is in turn a function of x.

Here y is the dependent variable, u is the intermediate variable, and x is the

independent variable. Then the “chain” of derivatives (chain rule) is as follows:

dy/dx = (dy/du) (du/dx)

Here dy/du is the first "chain link" and du/dx is the second "chain link". The
product of the two yields the derivative dy/dx.

With even more complicated functions it is helpful to define more than one intermediate

variable. i.e. y(x) = y[ u (w(x) ) ]

which reads as follows: y is a function of u, u in turn is a function of w, and w is in

in turn a function of x. Here y is the dependent variable, u is the first intermediate variable,

w is the second intermediate variable, and x is the independent variable.

Then the “chain” of derivatives (chain rule) is as follows:

dy/dx = (dy/du) (du/dw) (dw/dx)

So the derivative involves the product of "three chain links", dy/du, du/dw, dw/dx .

Note: It is usually up to you to select the intermediate variable or variables that simplify

calculation of the derivative, dy/dx.

Suppose y(x) = e^2x

Here y = dependent variable, x = independent variable, and say pick u = 2x as the

intermediate variable. Then y(u(x)) = e^u dy/du = e^u and du/dx = 2

Write the chain rule: dy/dx = (dy/du ) (du/dx) = 2 e^2x