Derivative
The rate of change of a quantity y with respect to another quantity x is called the derivative or differential coefficient of y with respect to x .
Differentiation of a Function
Let f ( x ) is a function differentiable in an interval [ a , b ]. That is , at every point of the interval , the derivative of the function exists finitely and is unique . Hence , we may define a new function g : [ a , b ] → R , such that , ∀ x ∈ [ a , b ], g ( x ) = f '( x ).
This new function is said to be differentiation ( differential coefficient ) of the function f ( x ) with respect to x and it is denoted by df ( x ) / d ( x ) or Df ( x ) or f '( x ).
Differentiation ‘ from First Principle
Let f ( x ) is a function finitely differentiable at every point on the real number line . Then , its derivative is given by
Standard Differentiations 1 . d / d ( x ) ( x n ) = nx n – 1 , x ∈ R , n ∈ R 2 . d / d ( x ) ( k ) = 0 , where k is constant .
3 . d / d ( x ) ( e x ) = e x 4 . d / d ( x ) ( a x ) = a x loge a > 0 , a ≠ 1