MEANING OF DERIVATIVES

The rate of change of y of a function with respect to the change of x of a function is called the derivative of the function. The derivative of a function f at a is defined by

                        f’(a) =h0[ f( a+h)- f(a)]h

The derivative of function f at any point x is given by 

                      f’(x)= df(x)dx= h0f(x+h)-f(x)h

MEANING OF DERIVATIVE-

Definition- 1:

If f is a real function and a is a point in its domain. So, the derivative of f at a can be defined as h0[ f( a+h)- f(a)]h  provided this limit exists. Derivative of f(x) at a is denoted by f’(a).

Here, f’(x) quantifies the change in f(x) at a with respect to x. 

Find the derivative of sinx at x=0.

Let f(x)= sinx . So, f'(0)=h0f(0+h) -f(0)h = sin(0+h) -sin (0)h   

                                             = h0sinhh= 1 

Definition-2:

Suppose f is a real-valued function, the function is defined by h0[ f(x+h )-f (x)]h wherever the limit exists is defined to be the derivative of f at x and is denoted by f’(x). This definition of derivative is said to be the first principle of derivative.

Thus, f’(x)= h0[f(x+h)-f(x)]h

The domain of f’(x) is wherever the above limit exists. There are different notations for the derivative of a function. f’(x) is represented by ddx(f (x)) or if y= f(x) is denoted by dydx. 

It is read as a derivative of f(x) or y with respect to x. It can be denoted as D(f(x)). Derivative of f at x=a is also denoted by ddxf(x) at x=a or dfdxx=a.

Let us take an example to understand the first principle of derivative.

Find the derivative of f(x)= sin x+cosx .

We know f’(x)= f(x+h)-f(x)h = h0sin(x+h)+cos(x+h)-sinx -cos xh

                                           = h0sin x cos h+ cosx sinh + cosx cosh -sinx sinh -sinx -cos x h

                                           = h0sin h(cos x-sinx) +sinx (cosh -1) +cosx (cosh -1)h

                                           = h0sinhh( cosx-sinx) +h0sinx cos h-1h+h0cosx cosh -1h

                                           = cosx-sinx

ALGEBRA OF  DERIVATIVE OF A FUNCTIONS- 

The derivative of a function is directly related to limits. The theorem are given below-

(i) Derivative of the sum of two functions is the sum of the derivatives of the functions. 

          ddx[ f(x) +g(x)]=ddxf(x) +ddxg(x)

(ii) Derivative of the difference of two functions is the difference of the derivatives of the functions.

              ddx[ f(x) -g(x)]= ddxf(x) – ddxg(x)

(iii) Derivative of the product of two functions is given by product rule.

             ddx[ f(x). g(x)] = ddxf(x). g(x) +f(x). ddxg(x) 

(iv) Derivative of the quotient of two functions is given by the quotient rule. 

             ddxf(x)g(x)= ddxf(x).g(x)-f(x).ddxg(x)[g(x)]2      

The last two theorems can be given in the other way also which helps them to recall easily. 

Let u = f(x) and v= g(x). Then, 

(uv)’= u’v +uv’

This is called Leibnitz rule. This rule is used for differentiating product of functions or in other words product rule. Similarly, the quotient rule is given by 

(uv)’= u’v-uv’v2

DERIVATIVE OF POLYNOMIALS AND TRIGONOMETRIC FUNCTIONS-

Here, we are going to discuss the theorem which tells us the derivative of the polynomial.

Let f(x) =anxn+ an-1xn-1+………+ a1x+ a0 be a polynomial function , where ais are all real numbers and an≠ 0. Then, the derivative function is given by

df(x)dx= nanxn-1+(n-1) an-1xx-2+ ……+ 2a2x + a1  

To understand this in a better way, let us discuss it with an example. 

 Find the derivative of  f(x)= 1+ x + x2 + x3+……+x50 at x=1. 

The derivative of above function is given by 1+ 2x +3x2+……+50x49. Calculating the value at  x=1,  the value of  this function equals 

1 +2(1) + 3(1)2 +……..+ 50(1)49= 1+ 2+……..+ 50 =(50)(51)2 = 1275

          

      STANDARD DERIVATIVES-

 

EXAMPLES-

1. Compute the derivative of tan x.

Let f(x)= tan x

ddxf(x) = h0f(x+h)- f(x)h = h0tan(x +h)- tanx h= h01hsin(x+h)cos(x+h)– sin xcos x

= h0sin(x+h) cos x- cos(x+h) sin xh cos(x+h)cos x= h0sin(x+h-x)h cos(x+h)cos x =  h0sinhh . h01cos(x+h)cos x

= 1. 1cos2x= sec2x

2. Compute the derivative of f(x)=sin2x.

  Using Leibnitz product rule, df(x)dx= ddx(sinx sinx)

                                                        = (sinx)’ sinx + sinx (sinx)’

                                                        = (cosx)sinx + sinx (cos x) = 2sinx cosx = sin2x