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
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
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
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
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
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