A derivative is a mathematical operation to find the rate of change of a function.
For a non linear function f(x) = y . The rate of change of f(x) correspond to change of x is equal to the ratio of change in f(x) over change in x
Then the Derivative of the function is defined as
but the derivative must exist uniquely at the point x. Seemingly well-behaved functions might not have derivatives at certain points. As examples, f(x)= 1/x has no derivative at x = 0; F(x) = |x| has two possible results at x = 0 (-1 for any value for which x<0 and +1 for any value for which x>0) On the other side, a function might have no value at x but a derivative of x, for example f(x)= x/x at x = 0. The function is undefined at x = 0, but the derivative is 0 at x = 0 as for any other value of x.
Practically all rules result, directly or indirectly, from a generalized treatment of the function.
Table of Derivative
Powers and Polynomials