Math Guide

Derivative ofthe Function

Derivative of the Function:


The slope of a curve at a single point is basically called the derivative. It is referred to as the slope of the tangent line or also instantaneous rate of change.


The definition of the derivative of a function f(x) can be given as, f’(x) = lim(h?0)[f(x+h) – f(x)]/h the condition being the limit exists. If the limit exists for each x in an open interval I then f(x) is said to be differentiable on I. A function f(x) is differentiable at x="a" only if f(x) has both right hand derivative and a left hand derivative at x="a" and also both the derivatives are equal. 


The right hand derivative of a function f(x) at x="a" in a domain ‘a’ can be given as,f’+(x) = lim(h?a+)[f(a+h) – f(a)]/h = lim(x?a+)[f(x) – f(a)]/(x-a)The left hand derivative of a function f(x) at x="a" in a domain ‘a’ can be given as, f’-(x) = lim(h?a-)[f(a+h) – f(a)]/h = lim(x?a-)[f(x) – f(a)]/(x-a)


 The conditions for the function f(x) to be differentiable on the interval ‘I’ are:If‘I’ has a right hand endpoint ‘a’, then the left hand derivative of the function f(x) exists at x="a"If ‘I’has a left hand endpoint ‘b’, then the right hand derivative of the function f(x) exists at x="b"Function f(x) is differentiable at all points of the interval ‘I’First Derivative of a functionThe first derivative of a function f(x) is the slope of the tangent line to the function at the point x and is represented as f’(x) or [f(x)]. 


It helps to determine whether the given function is increasing or decreasing and by how much. The graph of the function reflects this information by the slope of the function. 


As x increases f(x) also increases which is reflected as a positive slope and as x increases f(x) decreases which is reflected as a negative slope on the graph. 


 The following are the deductions of the first derivative of a the given function,If f’(a) > 0 then f(x) is increasing at x="a"If f’(a)<0 then f(x) is decreasing at x="a"If f’(a)=0 then x="a" is the critical point of the function f(x)For example, let f(x) =3x3 – 4x2+ x -1, the first derivative of f(x) = 9x2-8x+1; at x="0" f’(x) =1. This shows that f(x) is increasing at x="0" and at x="1," f’(x) =9(1)2-8(1) +1= 2. So, f(x) is an increasing function at x="1." Graph of a function and its DerivativeLet us graph the above function,Graph of f(x)= 3x3-4x2+x-1