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