u , v and w ; i.e., f = ( w o u ) o v . If t = v ( x ) and s = u ( t ), then ( o ) df d w u dw ds ds provided all the derivatives in the statement exist. Reader is invited to formulate chain rule for composite of more functions. Example Find the derivative of the function given by f ( x ) = sin ( x ).
Solution Observe that the given function is a composite of two functions. Indeed, if t = u ( x ) = x and v ( t ) = sin t , then f ( x ) = ( v o u ) ( x ) = v ( u ( x )) = v ( x ) = sin x Put t = u ( x ) = x . Observe that dv dt = and dx = exist. Hence, by chain rule df dv It is normal practice to express the final result only in terms of x .
Thus df cos EXERCISE . Differentiate the functions with respect to x in Exercises to . . sin ( x + ) .
cos (sin x ) . sin ( ax + b ) . sec (tan ( x )) . sin ( cos ( ax b cx .
cos x . sin ( x ) . cot x . .
Prove that the function f given by f ( x ) = | x – |, x ∈ R is not differentiable at x = . . Prove that the greatest integer function defined by f ( x ) = [ x ], < x < is not differentiable at x = and x = . .
. Derivatives of implicit functions Until now we have been differentiating various functions given in the form y = f ( x ). But it is not necessary that functions are always expressed in this form.