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CBSE Class 12 Maths Formula for Chapter-5 Continuity and Differentiability

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Continuity and Differentiability Formula for CBSE Class 12 Maths - Free PDF Download

When the board exams are near then the students will often feel pressure. As the exam gets closer one should be prepared with the subject and must have final revision notes with them. These notes not only include the one which you have written in the class but must include all the important topics of that particular chapter. Students often find it difficult to solve the problematic questions. Hence Vedantu teachers have prepared the list of these formulas, by using this one can easily solve the problems by following a certain method. In continuity and differentiability, all formulas are mentioned in the article.

Either it might be theory questions or problematic questions, going through this article can improve their confidence level to attend the exam. Some of the students might not prepare their revision notes, and when they start revising directly from the textbook or their class notes, then there are chances of missing out on one or two important topics when this happens then it is difficult to attend the questions that appear in the exam, hence to these type of students Vedantu has initiated to provide class 12 maths continuity and differentiability formulas. It is available on the official Vedantu website.

Continuity and differentiability are one of the most important topics which make the students understand some of the concepts such as continuity on an interval, continuity at a point, derivative of functions, and etc. ‘f’ is a real function that has point ‘c’ in its domain, then ‘f’ is said to be a continuous function if the value of the function at ‘c’, right-hand side limit and the left-hand side limit should be equal to each other. Let us see chapter 5 maths class 12 formulas.

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Continuity and Differentiability Class 12 Formulas

Let us see the conditions where the function is said to be continuous,

  • The real function ‘f’ is said to be left continuous at x = c, if there exists a function f(c), \[\lim_{x \rightarrow c^{-}} f(x)\] exists, and \[\lim_{x \rightarrow c^{-}} f(x) = f(c)\].

  • The real function ‘f’ is said to be left continuous at x = c, if there exists a function f(c),\[\lim_{x \rightarrow c^{+}} f(x)\] exists, and \[\lim_{x \rightarrow c^{+}} f(x) = f(c)\].

  • The real function ‘f’ is said to be continuous at x = c, if there exists a function f(c), \[\lim_{x \rightarrow c} f(x)\] exists, and \[\lim_{x \rightarrow c} f(x) = f(c)\].

In an open interval (a,b), a function ‘f’ is said to be continuous only if the function ‘f’ is continuous at every point in between that particular interval. In the closed interval [a,b] a function ‘f’ is said to be continuous along with that interval. If a function is continuous at a set of all the points then it is known as the domain of continuity. The domain of continuity is the proper subset of the domain of a function.


Properties and the Formula of Continuity and Differentiability Class 12:

  • Property 1: If the two functions f, g are said to be continuous at x = c, only if it satisfies all the conditions.

  1. α f is continuous at x = c, for all R.

  2. f + g is continuous at x = c.

  3. f - g is continuous at x = c.

  4. fg is continuous at x = c.

  5. f/g is continuous at x = c, where g(c) ≠ 0.

  • Property 2: D1 and D2 are said to be the domains of continuity of functions ‘f’ and ‘g’ respectively, it should satisfy the following functions,

  1. α f is continuous on D1, for all α ∈ R.

  2. f + g is continuous at D1D2.

  3. f - g is continuous at D1D2.

  4. fg is continuous at D1 D2.

  5. f/g is continuous at D1 D2, where g(c) ≠ 0.

  • Property 3: A polynomial function is said to be continuous everywhere.

  • Property 4: A rational function is said to be continuous in every point in its particular domain.

  • Property 5: If ‘f’ is said to be continuous at c, then |f| is continuous at x = c.

  • Property 6: If ‘f’ is said to be continuous one-one function that is defined in the closed interval [a, b] in a range [c, d] then, f-1:[c, d] ➝ [a, b].

  • Property 7: If ‘f’ is said to be continuous at c and ‘g’ is continuous at f(c) then gof is said to be continuous at c.

  • Property 8: All the basic trigonometric functions are continuous.

  • Property 9: All the inverse trigonometric functions are continuous

  • Property 10: Theorem, “If a function is continuous at any particular point, then it is necessarily continuous at that point”.

Till now we have studied about the properties and definitions of continuity and differentiability with formulas. Now let us learn the types of discontinuity, there are two main types of discontinuity:

  1. Removable discontinuity.

  2. Non-removable discontinuity.


All Formulas of Continuity and Differentiability Class 12

Derivative of a function ‘f’ is a real function and ‘c’ is a point in that domain then f at c is [f(c + h) - f(c)]/h it can be represented as f1(c) or  [d f(x)]/dx.


Some of the rules has to be followed to find the differentiation of a function:

  1. Algebra of Derivatives:

  • (u ± v)1 = u1 + v1

  • (uv)1 = u1v + v1u It is known as product rule.

  • (u/v)1 = [(u1v) - (v1u)]/v2 It is known as quotient rule.


  1. Chain Rule:

\[\frac{df}{dx} = \frac{dv}{du} \frac{du}{dx}\]


Along with the rules methods has to be followed, methods of continuity and differentiability formulas, are as follows,

  1. Parametric form: To differentiate y = g(t) and x = f(t) separately by ‘t’, dy/dx = (dy/dt) (dt/dx), using this result we can find dy/dx.

  2. Function of the Form g(x)f(x): If the function is in the form of y = g(x)f(x) and the f(x) and the g(x) is a differentiable function then log can be applied to obtain the form,

Log y = f(x) log g(x), this can be differentiated using product and chain rule.

  1. Implicit Function: If ‘y’ cannot be expressed as in terms of x, i.e as y = f(x) then this function is said to be an implicit function.

  2. Inverse trigonometric function.

  3. Derivative of Order Two and Three: The second derivative is the derivative form of first, it can be represented by f1, f2. The derivative of the second derivative is known as the third derivative.

  4. Mean Value Theorem: If f:[a, b] is continuous in the interval [a, b] and can be differentiable in (a, b) then f(a) = f(b), then f1(c) = 0.

  5. Rolle’s Theorem: If f:[a,b] is continuous in the interval [a, b] and can be differentiable in (a,b) then f(a) = f(b), then f1(c) = [f(b) - f(a)]/(b - a).


Continuity and Differentiability Class 12 Formulas

f(x)

f1(x)

xn

nxn-1

Sin x

Cos x

Cos x

-Sin x

Tan x

sec2x

Sec x

Sec x tan x

Cot x

- cosec2x

ex

ex

Log x

1/x

ax

axlog a


Conclusion:

Some of the important concepts covered in this chapter include differentiability, continuity, logarithmic functions, exponential functions, and mean value theorem. It is one of the important chapters in class 12 and it can be continued in higher classes also. For more information on class 12 maths chapter 5 all formulas visit the vedantu official website, and refer to the notes available in the website.

FAQs on CBSE Class 12 Maths Formula for Chapter-5 Continuity and Differentiability

1. Mention Continuity and Differentiability Class 12 all Formulas.

Ans: 

  • (u ± v)1 = u1 + v1

  • (uv)1 = u1v + v1u It is known as product rule.

  • (u/v)1 = [(u1v) - (v1u)]/v2 It is known as quotient rule.

  • Rolle’s Theorem: f1(c) = [f(b) - f(a)]/(b - a)

2. What is L Hospital Rule?

Ans: If functions f(x) and g(x) such that:

  • f(x) = 0

  • F(x) and g(x) are continuous functions at x = a.

  • F(x) and g(x) are continuous functions differentiable at x = a.

  • f1(x) and g1(x)are continuous at x = a.

To go through continuity and differentiability class 12 formulas pdf is available on the platform, hence one can access it easily and download it to refer whenever required.