Page 270 - DMTH201_Basic Mathematics-1

Unit 9: Differential Calculus

Notes
dy
Task Find , if,
dx

1 dy 1
1. If y 1 , prove that dx x y 2 1
x
1
x
x ........to

dy f x
2. If y f x f x f x ....... , prove that .
dx 2y 1

9.6 Summary

If x is a real variable, any expression in x is called a function of x. A function is denoted by
y = f(x), Where x is independent variable and y is dependent variable.

The functional value of f(x) at x = a is given by f(a).
If f(x) gets arbitrarily close to b (a finite number) for x sufficiently close to a, we say that f(x)
x
f
approaches the limit b as x approaches a, and write lim ( ) b
x a
lim ) x ( f ) x ( g lim f (x) lim g (x)
x a x a x a
lim x ( f ) x ( g ). lim f (x) . lim g (x)
x a x a
x a
A function y = f(x) is said to be differentiable at a point x = a, in its domain,
f (a h) - f (a)
lim exists.
h 0 h
Derivative of a constant is zero.

dy x
If y = e then e
x

dx
dy x
If y = a , then a log a
x
dx e
9.7 Keywords

Constant: A quantity whose value remains the same.

Function: A function ‘f’ from a set x to set y is a subset of x.y, denoted as {(x, y)}, such that
corresponding to each value of x, we can associated one and only one value of y. In such a
situation, y is said to be a function of x and is denoted as y = f(x).
Irrational Function: A function which is expressed as a root of a polynomial.
Parametric Function: If the variable x and y are given in terms of a new variable t, then the
function is said to be in the parametric form and ‘t’ is called the parameter.

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