First Principles

In this page first principles we are going to see some formula derivation.In order to find the derivative of a function y = f(x) from first-principle it is necessary to carry out the following operations.

1.Increase the argument x by Δx,calculate the increased value of the function y + Δy = f (x+ Δx)

2. Find the corresponding increment of the function Δy = f (x + Δx)- f(x)

3. From the ratio of the increment of the function to the increment of of the argument Δy/Δx = [f(x+Δx) -  f(x)]/Δx

4.Find the limit of this ratio Δx ---> 0;

dy/dx = f '(x) =    lim     [f(x+Δx) -  f(x)]/Δx

Δx ---> 0

We shall apply this general method for evaluating the derivatives of certain elementary functions.

Definition:

The derivative of a given function y = f(x) is defined as the limit of the ratio of the increment Δy of the function to the corresponding increment Δx  of the independent variable when the latter tends to zero.The symbols y' or f'(x) or dy/dx are used to denote derivative

Now we are going to see some example problems in this page.

1. The derivative of a constant function is zero.

2. The derivative x^n is nx^ (n-1),where n is a rational number.

3.The derivative of sin x is cos x

4. The derivative of cos x is - sin x

5. The derivative of logₐ x = 1/x log ₐ e

6. The derivative of logₑ x = 1/x

7.The derivative of tan x is sec² x

8. The derivative of sec x is sec x tan x

9. The derivative of cot x  is - cosec²x

10. The derivative of cosec x  is - cosec x cot x

11. The derivative of Sin -¹ x  is  1/√(1-x²)

12. The derivative of Cos -¹ x  is -1/√(1-x²)

13. The derivative of tan -¹ x  is 1/(1+x²)

14. The derivative of Cot -¹ x  is -1/(1+x²)

15. The derivative of Sec -¹ x  is  1/[x√(x² - 1)]

16. The derivative of cosec -¹ x  is -1/[x√(x² - 1)]

Related Topics  first principles

Quote on Mathematics

“Mathematics, without this we can do nothing in our life. Each and everything around us is math.

Math is not only solving problems and finding solutions and it is also doing many things in our day to day life.  They are:

It subtracts sadness and adds happiness in our life.

It divides sorrow and multiplies forgiveness and love.

Some people would not be able accept that the subject Math is easy to understand. That is because; they are unable to realize how the life is complicated. The problems in the subject Math are easier to solve than the problems in our real life. When we people are able to solve all the problems in the complicated life, why can we not solve the simple math problems?

Many people think that the subject math is always complicated and it exists to make things from simple to complicate. But the real existence of the subject math is to make things from complicate to simple.”

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