RD Sharma Class 12 Chapter 10 Solutions (Differentiability)
RD Sharma Class 12 Maths Solutions for Chapter 10 Differentiability are curated for you to prepare you for CBSE and competitive exams like JEE for higher studies. The book contains solutions to help you understand the concept from its foundation. The goal of these solutions is to make you apt in the Differentiability chapter and help you solve advanced problems with more ease. These RD Sharma Solutions are designed for students so that they understand various topics such as Differentiability at a point, Differentiability in a set, Definition and meaning of differentiability at a point, some results on differentiability, etc.
The solutions you find in RD Sharma Class 12 Maths Solutions Chapter 10 Differentiability comprise 2 exercises with a total of 20 questions. These questions will help you learn to utilize these concepts in mathematical problems that are explained in the Differentiability Chapter. You’ll see the problems and solutions are arranged in the same pattern as they are arranged in RD Sharma. Solutions curated here will benefit students to get the most out of the given amount of material in solving easy as well as an advanced level of problems they can expect in JEE Main as well as JEE Advanced.
If you study RD Sharma Class 12 Maths Solutions for Differentiability and solve these problems a number of times, it will definitely help you score good marks. These problems are solved in the easiest possible language for you to understand everything without any hurdles. Moreover, these solutions are presented by the most experienced subject matter experts with years of knowledge and practice in the same subject.
Topics Covered in RD Sharma Class 12 Maths Solutions for Chapter 10 Differentiability
Differentiability at a Point
If f(x) is a real-valued function on an open interval (a,b). Also, let c ∈ (a,b).
Then f(x) will be differentiable or will be a derivative at x = c,
if lim x→c [f(x)-f(c)/x-c]
You can call this limit a differential coefficient or a derivative of the function f(x) at point x=c.
You can denote this by f’(c) or (df(x)/dx) or Df(c).
Therefore, f’(c) = lim x→c [f(x)-f(c)/x-c]
Upon further calculation and methodology, you’ll also find the following hold true:
f(x) is differentiable at x =c ⇔ Lf’(c) = Rf’(c)
If Lf’(c) is not equal to Rf’(c), it can be said that f(x) is found not differentiable at x = c.
Meaning of differentiability at a point
Let the function f(x) be defined in (a,b) an open interval. And let P(c, f(c)) is a point on y = f(x), a curve. Also let the two neighbouring point Q(c-h, f(c-h)) and R(c+h, f(c+h)) found on the left as well as the right hand side of the P Point, then by further calculation and methodology,
f(x) will always be differentiable at the point x=c.
⇔ lim h->0 (f(c-h)-f(c))/-h = lim h->0 (f(c+h)-f(c)/h)
⇔At Point P, there is a slope of tangent that limits the chords’ position that is drawn on the left side of P and is found to be the same at the P’s slope of tangent that limits the chords’ position drawn on the right side of P.
⇔You may find the Point P bearing a unique tangent.
Theorem – Let a function f(x) be differentiable at x =c. It will always be continuous at x =c. However, the converse may not be necessarily true.
Differentiability in a Set
Let’s say a function f(x) is defined on (a,b), open interval. Then, it will be derivable or differentiable in (a,b), if it is found to be differentiable at all points of (a,b).
Some results on Differentiability
- All polynomial functions are differentiable at x ∈ R
- All constant functions are differentiable at x ∈ R.
- All logarithmic functions are differentiable at x ∈ R.
- All functions, trigonometric and inverse-trigonometric will always be differentiable in their domains.
- The difference, sum, quotient, and product of two functions that are differentiable is differentiable.
- Composition of a differentiable function will always be a differentiable function.
Discussion of Exercises of RD Sharma Solutions for Class 12 Chapter 10 Differentiability
- Exercise 10.1 discusses how continuity of a function at one point may not make it differentiable at the same point. It also lists questions where you’ll be required to prove Differentiability of a function at a given point or you will have to find the values of two variables in a given function so that the function is differentiable at a given point.
- Exercise 10.2 discusses finding the value of a variable in a given function. It may also ask for finding the derivative of a given function and discussing the continuity and discontinuity of a given function.
Benefits of RD Sharma Solutions for Class 12 Chapter 10 Differentiability by Instasolv
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