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First Principle Of Derivatives

First Principle of Derivatives Assignment / Homework Help
Application of the first principle of derivatives:

We have already seen that the derivative of a function y = f(x) by the first principle of differentiation is given by:

f ' (x) = dy
dx
= lim
h -> 0
f (x + h) - f(x)
h

In other words, we first find the difference quotient 
f (x + h) - f(x)
h
 and then find its limit as h -> 0.


Now let's apply this principle to find the derivatives of some elementary functions.

Find the derivative of the following functions by the first principle:

  • f(x) = c  (here c is a constant)

    f(x + h) = c

    f(x) = c

    Hence

    f (x + h) - f(x)
    h
    = c - c
    h
    = 0

    f ' (x) = lim
    h -> 0
    f (x + h) - f(x)
    h

    = lim
    h -> 0
    0

    = 0

    Therefore,
     d 
    dx
    (c) = 0

    From this we can conclude that the derivative of a constant function is a constant.

  • f(x) = xn, where n is a real number
    f(x + h) = (x + h)n
    f(x) = xn
    First Principle of Derivatives Assignment / Homework Help
    Hence, the difference quotient is given by:

    First Principle of Derivatives Assignment / Homework Help
    Finally, applying the limit we get

    First Principle of Derivatives Assignment / Homework Help

    Hence for f(x) = xn, f ' (x) = nxn-1

    In similar fashion we can derive the formulae for derivatives of standard functions listed below.

    First Principle of Derivatives Assignment / Homework Help First Principle of Derivatives Assignment / Homework Help

    We will apply the above formulae and basic differentiation rules (discussed in the next section) to find the derivative of any given function.

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