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# Xam Idea Class 12 Maths Chapter 6 Solutions: Continuity & Differentiability

Xam Idea Class 12 Maths Solutions for Chapter 6 ‘Continuity and Differentiability’ deals with questions based on the latest CBSE exam pattern for Class 12. These solutions will help you understand how to solve continuous equations when the values of their variable are changed. Continuity and Differentiability Solutions of Xam Idea Class 12 Maths Solutions will help you learn concepts like the sum, quotient, difference and product of continuous functions, every differentiable function can be continuous and it’s converse. Additionally, you will learn some fundamental theorems in this area like Rolle’s Theorem and Mean value theorem.

Xam Idea Class 12 Maths Solutions for Continuity and Differentiability consists of a combination of short and long questions. Specifically, there are 89  questions divided into 6 exercises consisting of Very Short Questions, Short Answer Questions and Long Answer Questions. The questions are based on the algebra of continuous functions, derivatives of composite functions, implicit functions, exponential and logarithmic functions and second-order derivatives. We have provided stepwise solutions for every exercise to help you in easy preparation for your Class 12 board exams.

Subject matter experts at Instasolv have prepared the solutions of Xam Idea Class 12 Maths book with the best possible methods. The solutions are not only based on the latest CBSE syllabus for Class 12 Maths but also provide a detailed understanding of the chapter. You can easily refer to 100% accurate solutions at any time and solve your doubts related to continuity and differentiability in minutes.

Important Topics for Xam Idea Class 12 Maths Solutions Chapter 6: Continuity and Differentiability

In a domain, if a limit of the function is equal to the value of the function at that point, then a real-valued function is continuous. Sum, quotient, difference and product of continuous functions are also continuous. For an instance, if a and b are continuous functions, then Differentiability

Finding derivatives of any function is a process of Differentiation. For the algebra of derivatives, three rules were established. Let’s say, ‘f’ is a real function and ‘a’ is a point at its domain,

So, the derivative of f at a is defined by Logarithmic Differentiation

When any equation is differentiated it should be noted that the variables must always be positive or else the logarithm is not possible to define.

Lets say, y=  f(x)= [ u(x)]v(x)

By taking it to the base e, log y= v(x) log  [ u(x)]

Using chain rule we get, Implicit Differentiation

When we differentiate each side of an equation with two variables and treat one of the variables as a function of the other variable, we are performing Implicit Differentiation. Usually, the two variables x and y are used.

Lets say x2+ y2 =1

By putting d / dx on both the sides we get, d / dx (x2+ y2 ) =  d / dx (1)

Therefore, d / dx (x2) +  d / dx( y2 ) =0

= 2x+2ydy / dx = 0

Thus,  dy / dx = – x / y

In the above example, we treated y as a function of x.

### Derivatives of Functions in Parametric form

Not all the times the variables are explicit or implicit. Sometimes it is also possible that the variable is linked with a third variable where it establishes an individual relationship between the first and second variable. This third variable is termed as a parameter.

Technically, when the relation between x and y are expressed in the form of x=f(t) and y=g(t), it is said to be a parametric form with ‘f’ as a parameter. So, by applying the chain rule we obtain, Mean Value Theorem

There are two conditions for mean value theorem

1. f (x) is continuous at a, b
2. f (x) is derivable at a, b

If the above conditions are satisfied, then it is said that there is some c in (a, b)

So, we get  f ‘ (c) = Rolle’s Theorem

In Rolle’s theorem, three conditions are essential to make the equation satisfied.

1. f (x) is continuous at a, b
2. f (x) is derivable at a, b
3. f (a)=f(b)

If the above three conditions are achieved, it is said that there exists some c in (a,b) making it f ’(c) = 0

### Exercise Discussion for Xam Idea Class 12 Maths Solutions Chapter 6: Continuity and Differentiability

In the exercise of the Xam Idea Class 12 Maths book Continuity and Differentiability, you will be solving the short and long problems related to the major topics. The questions are organized into various categories. They are as under

Here you will solve 7 questions based on finding the derivatives. You are bound to give answers in two to three lines.

The questions are related to differentiability and you will be solving 11 questions under this category. You are allowed to give answers in 3 to 4 lines.

You need to find the value of k and discuss the continuity and differentiability in this exercise. Apparently, you will witness 16 questions. Here you can long enough to determine the given problem.

This exercise is full of questions where you need to prove the equations and differentiate various functions. It consists of 36 questions.

This exercise has a whole lot of 15 PYQ (Previous Year Questions) based on proving equations which can be solved with the help of differentiating them.