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of Derivatives:
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:
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 -> 0f (x + h) - f(x)
h
= lim
h -> 00
= 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
Hence, the difference quotient is given by:
Finally, applying the limit we get
Hence for f(x) = xn, f ' (x) = nxn-1
In similar fashion we can derive the formulae for derivatives of standard functions listed below.
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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