| is a continuous function. Consider the left hand limit – ( ( ) f f = − The right hand limit ( ( ) f f Since the above left and right hand limits at are not equal, ( ( ) f f does not exist and hence f is not differentiable at . Thus f is not a differentiable function. .
. Derivatives of composite functions To study derivative of composite functions, we start with an illustrative example. Say, we want to find the derivative of f , where f ( x ) = ( x + ) One way is to expand ( x + ) using binomial theorem and find the derivative as a polynomial function as illustrated below. d f x ( ) ( ) = x + x + = ( x + ) Now, observe that f ( x ) = ( h o g ) ( x ) where g ( x ) = x + and h ( x ) = x .
Put t = g ( x ) = x + . Then f ( x ) = h ( t ) = t . Thus df ( x + ) = ( x + ) . = t .
= dh dt The advantage with such observation is that it simplifies the calculation in finding the derivative of, say, ( x + ) . We may formalise this observation in the following theorem called the chain rule. Theorem (Chain Rule) Let f be a real valued function which is a composite of two functions u and v ; i.e., f = v o u . Suppose t = u ( x ) and if both dt dx and dv dt exist, we have df dv dt dt dx We skip the proof of this theorem.
Chain rule may be extended as follows. Suppose f is a real valued function which is a composite of three functions