Calculus is a wide branch of mathematics. It has mainly two parts, one is differentiation and the other one is Integration. Both the parts have their own significance in practical life.

As we know that the process of finding the derivative is known as differentiation. It is denoted by d(x)/dx . And fundamentally it is calculated by the first principle of differentiation.

There are some standard results of differentiation which can be tabulated below.

1. d(xn)/dx = n xn-1

2. d(ex)/dx = ex

3. d(ax)/dx = ax log a

4. d(log x)/dx = 1/x

5. d(sin x)/dx = cos x

6. d(cos x)/dx = - sin x

7. d(tan x)/dx = sec2x

8. d(cot x)/dx = - cosec2 x

9. d(sec x)/dx= secx tanx

10.d(cosec x)/dx = - cosecx cotx

11.d(sin-1 x)/dx = 1/ (1-x2)1/2

12.d(cos-1 x)/dx = -1/(1-x2)1/2

13.d(tan-1 x)/dx = 1/ (1+x2)

14.d(sec-1 x)/dx = 1/(x2 -1)1/2(x)

15.d(cot-1 x)/dx = -1 /(1+x2)

Thus with the help of above formulas we can solve more difficult problems of differentiation.

Now there are some fundamental theorems of differentiation which help to solve the problems .they are as below

· Fundamental theorem of differentiation

1. Derivative Of Fractions Quotient Rule

To find the derivative of quotient of two Functions let there are two functions u and v which are function of x.

So

Y = u/v

Y+δy = u+δu/v+δv

δy = (u+δu)v – u(v+δv)/( v+δv)v

δy/δx = v (δu/δx)-u(δv/δx) / (v+δv)v

lim δx→0 δy/δx = lim δx→0 [v (δu/δx)-u(δv/δx)/ (v+δv)v]

lim δx→0 δy/δx = v( lim δx→0 δu/δx) – u (lim δx→0 δv/δx)/v+ lim δx→0 (δv/δx) lim δx→0 δx *v

dy/dx = v (du/dx) – u (dv/dx)/v2

d(u/v)/dx = v (du/dx) – u (dv/dx)/v2

This is the basic formula to calculate the differentiation by the quotient rule. This formula is used directly in rare case but it is used in the middle part of the question to solve the problem further.

There are many other theorems also which are used to solve the problem like multiplication theorem, addition /subtraction theorem etc.