If y is a function of u , u is a function of v and v is a function of x . Then , dy / dx = dy / du * du / dv * dv / dx .
2 . Differentiation Using Substitution In order to find differential coefficients of complicated expression involving inverse trigonometric functions some substitutions are very helpful , which are listed below .
3 . Differentiation of Implicit Functions
If f ( x , y ) = 0 , differentiate with respect to x and collect the terms containing dy / dx at one side and find dy / dx .
Shortcut for Implicit Functions For Implicit function , put d / dx { f ( x , y )} = – ∂f / ∂x / ∂f / ∂y , where ∂f / ∂x is a partial differential of given function with respect to x and ∂f / ∂y means
Partial differential of given function with respect to y . 4 . Differentiation of Parametric Functions If x = f ( t ), y = g ( t ), where t is parameter , then dy / dx = ( dy / dt ) / ( dx / dt ) = d / dt g ( t ) / d / dt f ( t ) = g ’ ( t ) / f ’ ( t )