Summary of Class 12 Maths Continuity and Differentiability
Continuity of a Function
The real valued function is said to be a continuous,
if it is continuous at all points in its domain.
Suppose f
is a real valued function on a subset of the real numbers and let c be
a point in the domain of f. Then, f is continuous at c if its left hand limit, right hand limit and the value of the function
are equal.
i.e., 
= 
= f(c).
Algebra of Continuous Functions
Let f and g be
two real functions continuous at a real number c, then
(i) (f + g)
is continuous at x = c.
(ii) (f – g)
is continuous at x = c.
(iii) f . g is
continuous at x = c.
(iv) (f/g) is
continuous at x = c, provided g(x) ≠ 0.
Differentiability of a Function
Suppose f is a
real function and c is a point in its domain. The derivative of f at
c is defined by 
provided this limit
exists.
The derivative of f at
c is denoted by f ′(c) or [
f(x)]x
= c
The derivative of f is defined by 
wherever the limit exists, is defined to
be the derivative of f.
The derivative of f is
denoted by f ′(x) or f(x). If y
= f(x), then the derivative is denoted by 
or y’.
The process of finding derivative
of a function is called differentiation.
Theorem
If a function f is
differentiable at a point c, then it is also continuous at that point.
Thus, every differentiable function is continuous, but the converse is not
true.
Chain Rule
Chain rule is used to differentiate a composite function, which is a function within
another function.
If y = f(u)
and u = g(x), then the chain rule can be expressed as:
dy/dx = (dy/du).(du/dx)
Or, for a composite function f(g(x)): d/dx [f(g(x))] = f'(g(x)).g'(x)
Algebra of Derivatives
Let u and v be functions of x. Then,
(i) (u ± v) = 
± 
(Sum and difference
rule)
(ii) (u.v) = u + v (Product rule)
(iii) (u/v) = v - u / v2 (Quotient rule)
Derivatives of Some Standard Functions
Derivatives of Implicit Functions
Let f(x, y)
= 0 be an implicit function of x and y. To find , differentiate both side of the equation w.r.t. x and then
shift all the terms involving on LHS and remaining
terms on RHS to find the required value.
Derivatives of Inverse Trigonometric Functions
Differentiation of Logarithmic Functions
Let the given function is of the form y = [u(x)]v(x),
where u(x) and v(x) are functions of x. In such cases, we take logarithm
on both sides and use properties of logarithm to simplify it and then
differentiate it to find .
Derivatives of Functions in Parametric Forms
Let the given
parametric equations are:
x = f(t) and y = g(t),
where t is called the parameter.
Then, = 
/ 
Second Order Derivative
Let y = f(x) be a function. Then, = f’(x) is called the first
derivative of y or f(x).
Differentiating it again, we get () = [f’(x)] => 
= f’’(x)
is called the second order derivative of y
w.r.t. x.
It is also denoted by D2 y or y’’ or y2.