Continuity And Differentiability Class 12 Notes Maths Chapter 5 - CBSE


What are Continuity And Differentiability ?


A real valued function is continuous at a point in its domain if the limit of the function at that point equal the value of the function at that point. A function f is continous at an interior point x = c of its domain if limx→cf(x) = f(c).


Suppose f(x) is a real valued function and x=c is a point in its domain. The derivative of f(x) at x=c is defined by limx→c( f(x)-f(c))/x-c or limh→0( f(c+h)-f(c))/h provided this limit exists. Then the function defined by limh→0( f(x+h)-f(x))/h wherever the limit exists, is defined to be derivative of f(x). The process of finding the derivative of a function is called differentiation w.r.t x.

Chain Rule

Let f be a real valued function which is a composite of two functions u and v; i.e. f = vou. Suppose t = u(x) and if both dt/dx and dv/dx exists, we ave,df/dx=dv/dt.dt/dx

Derivative Of Inverse Trigonometric Function

f(x) sin-1 x cos-1 x tan-1 x cot-1 x sec-1 x cosec-1 x
f'(x) $$\frac{1}{\sqrt[\normalsize]{1-x^2}}$$ $$\frac{-1}{\sqrt[\normalsize]{1-x^2}}$$ $$\frac{1}{1+x^2}$$ $$\frac{-1}{1+x^2}$$ $$\frac{1}{\sqrt[\normalsize{x}]{1-x^2}}$$ $$\frac{-1}{\sqrt[\normalsize{x}]{1-x^2}}$$

Derivative Of Implicit Functions

If the variables x and y are connected by the relation of the form f(x,y) = 0 and it is not possible to express y as a function of x in the form y = φ(x), then y is called an implicit function of x. In such case, we differentiate dφ/dy.dy/dx

Logarithmic Differentiation

Logarithmic differentiation is a powerful technique to differentiate functions of the form f(x) = [u(x)]v(x). Here both f(x) and u(x) need to be positive for this technique to make sense.

Derivative Of Functions Expressed In Parametric Forms

If x = φ(t) and y = φ(t) are two functions and t is a variable. In such cases x and y are called parametric functions and t is called the parameter. By chain rule we get; dy/dx=φ'(t)/φ'(t)

Second Order Derivatives

If y = f(x), then dy/dx is the derivative of y w.r.t x. Then we shall call dy/dx as the first order derivative of y w.r.t x and the derivative of dy/dx w.r.t x as the second order derivative d2y/dx2 of y w.r.t x.