What is Differentiation? – Derivative basics

Differentiation – commonly known as “Derivative”, is the most importantly, it is basic & essential operation in Mathematics. Differentiation gives the resultant of the small change of dependent quantity corresponding to the small change in its independent quantity. In this post, we’ll discuss firstly the definition & after that the working process of the Derivatives..

Differentiation –

Derivative is a function that gives the resultant of change of dependent value in respect of the independent value. In other words, we can say, that it gives, the rate of change of the dependent function. The definition of derivative comes as a result of limit of the function. so, we can define it as follows-

Furthermore, derivative is – difference of changed & the original value of dependent divides the difference in changed & the original value of independent value.

Definition:

Let, y=f(x) be a function of x & δy is the be a change in y, as a result of small change in x.

then, to find derivative-  limδx→∞ (δy/δx)= limδx→∞ (f(x-δx)-f(δx))/δx

is the derivative of y with respect to x, which is consequently given as- dy/dx or d/dx(f(x)).

Thus,similarly –

dy/dx = limδx→∞ (f(x-δx)-f(δx))/δx


Working of Derivatives:

Derivative of a function most importantly proceeds along with the standard results , One must get these results in order to solve differentiation problems with ease.

Furthermore, the results consecutively for linear, logarithmic/exponential also for trigonometric functions are as follows :

Linear Functions-

  1. d/dx (Xn) = n.(Xn-1)
  2. d/dx (1/Xn) =-n/(Xn+1)
  3. d/dx (1/√X) =1/(2√X)
  4. d/dx (X) =1

Exponential/Logarithmic Functions-

  1. d/dx (ax) = (ax. log(a))
  2. d/dx (ex) = (ex)
  3. d/dx (logxa) =-n/(1/ x . log a)
  4. d/dx (log x) =1/(X)

Trigonometric Functions-

  1. d/dx (sin x) =  cos x
  2. d/dx (cos x) = -sin x
  3. d/dx (tan x) = sec 2 x
  4. d/dx (cot x) =- cosec 2 x
  5. d/dx (sec x) = sec x . tan x
  6. d/dx (cosec x) =- cosec x . cot x
  7. d/dx (constant) = 0

This post talks about the basics in differentiation. However, we’ll discuss the properties in addition the rules of derivatives also its examples accordingly in the further posts..