unit 1 differential calculus - IGNOU

1.6.1 Derivative of a Function
We now define the derivative of a given function by using the notion of limit.
Definition 10
The function y = f (x) is said to have a derivative (or be differentiable) at a
point x if, and only if, the limit
lim
h →0
f ( x + h)
− f ( x)
h
dy
exists and is finite. This limit, written as or f ′ (x), is called the
dx
derivative of f (x) with respect to x.
If the derivative of a function exists at every point of an interval, then we say that
the function is differentiable in that interval. However, while considering the
derivatives at the end points of an interval, we evaluate suitable one-sided
derivatives. For example, if the interval is [a, b], then the derivative at x = a is
calculated as
lim
+
h →0
f ( a + h)
− f ( a)
and at b as
h
lim
−
h →0
f ( b + h)
− f ( b)
h
If f ′ (x) is continuous at a point, then we say that f (x) is continuously
differentiable at that point.
Example 1.21
Solution
Let y = x n (n is a positive integer). Prove that .
1 − dy n
= nx
dx
Then
y = f ( x)
=
n
x
n
f ( x + h)
− f ( x)
( x + h)
− x
=
h
h
( x + h − x)
−1
n −
=
[ ( x + h)
+ ( x + h)
h
n 2
The above step is derived by using the result
We now have
f
a
n
− b
n
= ( a − b)
[ a
x
n
+
. . .
+ ( x + h)
x
b + . . . + ab
n − 2 n −1
+ x
n −1
n − 2
n − 2 n −1
+ a
+ b
+ −
n
= ( x + h)
+ ( x + h)
x + . . . + ( x + h)
x + x
h
( x h)
f ( x)
n −1
n − 2
n − 2 −1
Taking the limit h → 0, we get
dy
dx
=
lim
h→
0
f ( x + h)
− f ( x)
= x
h
Therefore, .
1 − dy n
= nx
dx
n −1
+ x
n − 2
x +
. . .
+ x
n −1
(n number of terms)
]
]
Differential Calculus
37