Chain Rule Formulas

There are two forms of chain rule formula as shown below.

Chain Rule Formula 1 

d/dx ( f(g(x) ) = f’ (g(x)) · g’ (x)

Example:  To find the derivative of d/dx .(log5x),
       f(g(x))=log5x
      where, f(x) = log(x) and, g(x) = 5x
       By chain rule,
       d/dx .(log5x) = 5/5x = 1/x.

Chain Rule Formula 2

dy/dx = dy/du · du/dx

Example: To find the derivative of  d/dx .(log5x),

Suppose, y=log5x, and u=5x.
By the chain rule formula,
d/dx. (log5x) = d/du .(log5x) . du/dx
              =1/5x .5
d/dx. (log5x) =1/x.

Double Chain Rule

Functions that depend on multiple variables may be nested one on top of the other. To obtain the total derivative, the series of smaller derivatives are multiplied together. Let’s say there are 3 functions: u, v, and w. A function is a combination of u, v, and w. Here, the chain rule is broadened. The chain rule is used twice when a function is made up of three different functions. 
If f = (u o v) o w = df/dx = df/du. du/dv. dv/dw. dw/dx

Example1:  y = (1+tan3x)²

                        y’ = 6(1+tan3x). sec²3x

Example 2: y = (2x-5)²

                                y’ = 4(2x-5)

The chain rule is an important topic of Quantitative Aptitude that needs to be practiced well for competitive exams. The following article includes the concepts, steps, and formulas that are used to solve the chain rule questions.