Commerce Exam > Commerce Notes > Mathematics (Maths) Class 11 > NCERT Solutions Miscellaneous Exercise 2: Limits and Derivatives
NCERT Solutions Class 11 Maths Chapter 12 - Limits and Derivatives
| Download, print and study this document offline |
Please wait while the PDF view is loading
Page 1 Class XI Chapter 13 – Limits and Derivatives Maths ______________________________________________________________________________ Question 21: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): ? ? sin cos xa x ? Solution 21: Let f(x) = ? ? sin cos xa x ? By quotient rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 2 2 cos sin sin cos ' cos cos sin sin sin ' ... cos dd x x a x a x dx dx fx x d x x a x a x dx f x i x ? ? ? ?? ?? ? ? ? ? ? ?? ?? ? Let g(x)=sin (x + a). Accordingly, g(x + h) = sin (x + h + a) By first principle, ? ? ? ? ? ? ? ? 0 0 0 0 0 () ' lim 1 lim sin sin 1 lim 2cos sin 22 1 2 2 lim 2cos sin 2 sin 22 2 lim cos 2 h h h h h g x h g x gx h x h a x a h x h a x a x h a x a h x a h h hh h x a h h h ? ? ? ? ? ?? ? ? ? ? ? ? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? ?? ?? ?? ?? ???? ?? ?? ? ?? ?? ?? ?? ? ?? ? ?? ? ? ? ? ? 0 0 2 0 sin 2 2 h 2 limcos .lim As h 0 0 2 2 2 2 sinh cos 1 lim 1 2 cos ... h h h h x a h h h xa h x a ii ? ? ? ?? ?? ?? ?? ? ?? ? ? ?? ?? ?? ?? ?? ?? ?? ? ? ? ? ?? ? ? ? ? ?? ?? ?? ?? ? ? ? ? ?? ?? ?? ?? ?? ? ? ? ? ? ? ? ? ?? ?? ? ? ? ? ?? From (i) and (ii), we obtain Page 2 Class XI Chapter 13 – Limits and Derivatives Maths ______________________________________________________________________________ Question 21: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): ? ? sin cos xa x ? Solution 21: Let f(x) = ? ? sin cos xa x ? By quotient rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 2 2 cos sin sin cos ' cos cos sin sin sin ' ... cos dd x x a x a x dx dx fx x d x x a x a x dx f x i x ? ? ? ?? ?? ? ? ? ? ? ?? ?? ? Let g(x)=sin (x + a). Accordingly, g(x + h) = sin (x + h + a) By first principle, ? ? ? ? ? ? ? ? 0 0 0 0 0 () ' lim 1 lim sin sin 1 lim 2cos sin 22 1 2 2 lim 2cos sin 2 sin 22 2 lim cos 2 h h h h h g x h g x gx h x h a x a h x h a x a x h a x a h x a h h hh h x a h h h ? ? ? ? ? ?? ? ? ? ? ? ? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? ?? ?? ?? ?? ???? ?? ?? ? ?? ?? ?? ?? ? ?? ? ?? ? ? ? ? ? 0 0 2 0 sin 2 2 h 2 limcos .lim As h 0 0 2 2 2 2 sinh cos 1 lim 1 2 cos ... h h h h x a h h h xa h x a ii ? ? ? ?? ?? ?? ?? ? ?? ? ? ?? ?? ?? ?? ?? ?? ?? ? ? ? ? ?? ? ? ? ? ?? ?? ?? ?? ? ? ? ? ?? ?? ?? ?? ?? ? ? ? ? ? ? ? ? ?? ?? ? ? ? ? ?? From (i) and (ii), we obtain Class XI Chapter 13 – Limits and Derivatives Maths ______________________________________________________________________________ ? ? ? ? ? ? ? ? 2 2 2 cos .cos sin sin ' cos cos cos cos cos x x a x x a fx x x a x x a x ? ? ? ? ?? ? ? Question 22: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): x 4 (5 sin x - 3 cos x) Solution 22: Let 4 ( ) (5sin 3cos ) f x x x x ?? By product rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? 44 44 43 3 '( ) (5sin 3cos ) (5sin 3cos ) 5 sin 3 cos (5sin 3cos ) 5cos 3 sin (5sin 3cos ) 4 5 cos 3 sin 20sin 12cos dd f x x x x x x x dx dx d d d x x x x x x dx dx dx x x x x x x x x x x x x x ? ? ? ? ?? ? ? ? ? ?? ?? ? ? ? ? ? ?? ?? ? ? ? ? Question 23: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (x 2 + 1) cos x Solution 23: Let f(x) = ? ? 2 1 cos xx ? By product rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? 22 2 2 ' 1 cos cos 1 1 sin cos 2 sin sin 2 cos dd f x x x x x dx dx x x x x x x x x x ? ? ? ? ? ? ? ? ? ? ? ? Page 3 Class XI Chapter 13 – Limits and Derivatives Maths ______________________________________________________________________________ Question 21: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): ? ? sin cos xa x ? Solution 21: Let f(x) = ? ? sin cos xa x ? By quotient rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 2 2 cos sin sin cos ' cos cos sin sin sin ' ... cos dd x x a x a x dx dx fx x d x x a x a x dx f x i x ? ? ? ?? ?? ? ? ? ? ? ?? ?? ? Let g(x)=sin (x + a). Accordingly, g(x + h) = sin (x + h + a) By first principle, ? ? ? ? ? ? ? ? 0 0 0 0 0 () ' lim 1 lim sin sin 1 lim 2cos sin 22 1 2 2 lim 2cos sin 2 sin 22 2 lim cos 2 h h h h h g x h g x gx h x h a x a h x h a x a x h a x a h x a h h hh h x a h h h ? ? ? ? ? ?? ? ? ? ? ? ? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? ?? ?? ?? ?? ???? ?? ?? ? ?? ?? ?? ?? ? ?? ? ?? ? ? ? ? ? 0 0 2 0 sin 2 2 h 2 limcos .lim As h 0 0 2 2 2 2 sinh cos 1 lim 1 2 cos ... h h h h x a h h h xa h x a ii ? ? ? ?? ?? ?? ?? ? ?? ? ? ?? ?? ?? ?? ?? ?? ?? ? ? ? ? ?? ? ? ? ? ?? ?? ?? ?? ? ? ? ? ?? ?? ?? ?? ?? ? ? ? ? ? ? ? ? ?? ?? ? ? ? ? ?? From (i) and (ii), we obtain Class XI Chapter 13 – Limits and Derivatives Maths ______________________________________________________________________________ ? ? ? ? ? ? ? ? 2 2 2 cos .cos sin sin ' cos cos cos cos cos x x a x x a fx x x a x x a x ? ? ? ? ?? ? ? Question 22: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): x 4 (5 sin x - 3 cos x) Solution 22: Let 4 ( ) (5sin 3cos ) f x x x x ?? By product rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? 44 44 43 3 '( ) (5sin 3cos ) (5sin 3cos ) 5 sin 3 cos (5sin 3cos ) 5cos 3 sin (5sin 3cos ) 4 5 cos 3 sin 20sin 12cos dd f x x x x x x x dx dx d d d x x x x x x dx dx dx x x x x x x x x x x x x x ? ? ? ? ?? ? ? ? ? ?? ?? ? ? ? ? ? ?? ?? ? ? ? ? Question 23: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (x 2 + 1) cos x Solution 23: Let f(x) = ? ? 2 1 cos xx ? By product rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? 22 2 2 ' 1 cos cos 1 1 sin cos 2 sin sin 2 cos dd f x x x x x dx dx x x x x x x x x x ? ? ? ? ? ? ? ? ? ? ? ? Class XI Chapter 13 – Limits and Derivatives Maths ______________________________________________________________________________ Question 24: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (ax 2 + sin x) (p + q cos x) Solution 24: Let ? ? ? ? 2 sin f x ax x ?? (p + q cos x) By product rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 22 2 2 ' sin cos cos sin sin sin cos 2 cos sin sin cos 2 cos dd f x ax x p q x p q x ax x dx dx ax x q x p q x ax x q x ax x p q x ax x ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Question 25: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (x + cos x) (x – tan x) Solution 25: Let f(x) = (x + cos x) (x – tan x) By product rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ' (x + cos x) tan tan cos x + cos x tan tan 1 sin x + cos x 1 tan tan 1 sin ... Let g x tan . Accordingly, g(x+h)=tan x+h dd f x x x x x x x dx dx dd x x x x x dx dx d x x x x i dx x ? ? ? ? ? ?? ? ? ? ? ? ?? ?? ?? ? ? ? ? ? ?? ?? ? By first principle, ? ? ? ? ? ? ? ? ? ? 0 0 0 ' lim tan tan lim 1 sin( ) sin lim cos cos h h h g x h g x gx h x h x h x h x h x h x ? ? ? ?? ? ?? ? ?? ? ?? ?? ? ?? Page 4 Class XI Chapter 13 – Limits and Derivatives Maths ______________________________________________________________________________ Question 21: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): ? ? sin cos xa x ? Solution 21: Let f(x) = ? ? sin cos xa x ? By quotient rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 2 2 cos sin sin cos ' cos cos sin sin sin ' ... cos dd x x a x a x dx dx fx x d x x a x a x dx f x i x ? ? ? ?? ?? ? ? ? ? ? ?? ?? ? Let g(x)=sin (x + a). Accordingly, g(x + h) = sin (x + h + a) By first principle, ? ? ? ? ? ? ? ? 0 0 0 0 0 () ' lim 1 lim sin sin 1 lim 2cos sin 22 1 2 2 lim 2cos sin 2 sin 22 2 lim cos 2 h h h h h g x h g x gx h x h a x a h x h a x a x h a x a h x a h h hh h x a h h h ? ? ? ? ? ?? ? ? ? ? ? ? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? ?? ?? ?? ?? ???? ?? ?? ? ?? ?? ?? ?? ? ?? ? ?? ? ? ? ? ? 0 0 2 0 sin 2 2 h 2 limcos .lim As h 0 0 2 2 2 2 sinh cos 1 lim 1 2 cos ... h h h h x a h h h xa h x a ii ? ? ? ?? ?? ?? ?? ? ?? ? ? ?? ?? ?? ?? ?? ?? ?? ? ? ? ? ?? ? ? ? ? ?? ?? ?? ?? ? ? ? ? ?? ?? ?? ?? ?? ? ? ? ? ? ? ? ? ?? ?? ? ? ? ? ?? From (i) and (ii), we obtain Class XI Chapter 13 – Limits and Derivatives Maths ______________________________________________________________________________ ? ? ? ? ? ? ? ? 2 2 2 cos .cos sin sin ' cos cos cos cos cos x x a x x a fx x x a x x a x ? ? ? ? ?? ? ? Question 22: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): x 4 (5 sin x - 3 cos x) Solution 22: Let 4 ( ) (5sin 3cos ) f x x x x ?? By product rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? 44 44 43 3 '( ) (5sin 3cos ) (5sin 3cos ) 5 sin 3 cos (5sin 3cos ) 5cos 3 sin (5sin 3cos ) 4 5 cos 3 sin 20sin 12cos dd f x x x x x x x dx dx d d d x x x x x x dx dx dx x x x x x x x x x x x x x ? ? ? ? ?? ? ? ? ? ?? ?? ? ? ? ? ? ?? ?? ? ? ? ? Question 23: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (x 2 + 1) cos x Solution 23: Let f(x) = ? ? 2 1 cos xx ? By product rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? 22 2 2 ' 1 cos cos 1 1 sin cos 2 sin sin 2 cos dd f x x x x x dx dx x x x x x x x x x ? ? ? ? ? ? ? ? ? ? ? ? Class XI Chapter 13 – Limits and Derivatives Maths ______________________________________________________________________________ Question 24: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (ax 2 + sin x) (p + q cos x) Solution 24: Let ? ? ? ? 2 sin f x ax x ?? (p + q cos x) By product rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 22 2 2 ' sin cos cos sin sin sin cos 2 cos sin sin cos 2 cos dd f x ax x p q x p q x ax x dx dx ax x q x p q x ax x q x ax x p q x ax x ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Question 25: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (x + cos x) (x – tan x) Solution 25: Let f(x) = (x + cos x) (x – tan x) By product rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ' (x + cos x) tan tan cos x + cos x tan tan 1 sin x + cos x 1 tan tan 1 sin ... Let g x tan . Accordingly, g(x+h)=tan x+h dd f x x x x x x x dx dx dd x x x x x dx dx d x x x x i dx x ? ? ? ? ? ?? ? ? ? ? ? ?? ?? ?? ? ? ? ? ? ?? ?? ? By first principle, ? ? ? ? ? ? ? ? ? ? 0 0 0 ' lim tan tan lim 1 sin( ) sin lim cos cos h h h g x h g x gx h x h x h x h x h x h x ? ? ? ?? ? ?? ? ?? ? ?? ?? ? ?? Class XI Chapter 13 – Limits and Derivatives Maths ______________________________________________________________________________ ? ? ? ? 0 sin( )cos sin cos 1 lim cos cos h x h x x x h h x h x ? ?? ? ? ? ? ?? ? ?? ? ? ? ? ? ? ? ? ? ? 0 0 00 2 2 1 1 in( ) lim cos cos 1 1 sinh lim cos cos 1 sinh 1 . lim . lim cos cos 11 .1. cos cos 0 1 cos sec h h hh s x h x x h x h x h x h x h x h xx x x ii ? ? ?? ?? ?? ? ?? ? ?? ?? ? ?? ? ?? ?? ?? ? ?? ?? ?? ? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 2 2 2 Therefore, from (i) and ii ,We obtain f' x cos (1 sec ) tan 1 sin cos tan tan 1 sin tan cos tan 1 sin x x x x x x x x x x x x x x x x x x ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Question 26: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non zero constants and m and n are integers): 4 5sin 3 7cos xx xx ? ? Solution 26: Let f(x) = 4 5sin 3 7cos xx xx ? ? By quotient rule, ? ? ? ? ? ? ? ? ? ? ? ? 2 3 7cos 4 5sin 4 5sin 3 7cos ' 3 7cos dd x x x x x x x x dx dx fx xx ? ? ? ? ? ? ? Page 5 Class XI Chapter 13 – Limits and Derivatives Maths ______________________________________________________________________________ Question 21: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): ? ? sin cos xa x ? Solution 21: Let f(x) = ? ? sin cos xa x ? By quotient rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 2 2 cos sin sin cos ' cos cos sin sin sin ' ... cos dd x x a x a x dx dx fx x d x x a x a x dx f x i x ? ? ? ?? ?? ? ? ? ? ? ?? ?? ? Let g(x)=sin (x + a). Accordingly, g(x + h) = sin (x + h + a) By first principle, ? ? ? ? ? ? ? ? 0 0 0 0 0 () ' lim 1 lim sin sin 1 lim 2cos sin 22 1 2 2 lim 2cos sin 2 sin 22 2 lim cos 2 h h h h h g x h g x gx h x h a x a h x h a x a x h a x a h x a h h hh h x a h h h ? ? ? ? ? ?? ? ? ? ? ? ? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? ?? ?? ?? ?? ???? ?? ?? ? ?? ?? ?? ?? ? ?? ? ?? ? ? ? ? ? 0 0 2 0 sin 2 2 h 2 limcos .lim As h 0 0 2 2 2 2 sinh cos 1 lim 1 2 cos ... h h h h x a h h h xa h x a ii ? ? ? ?? ?? ?? ?? ? ?? ? ? ?? ?? ?? ?? ?? ?? ?? ? ? ? ? ?? ? ? ? ? ?? ?? ?? ?? ? ? ? ? ?? ?? ?? ?? ?? ? ? ? ? ? ? ? ? ?? ?? ? ? ? ? ?? From (i) and (ii), we obtain Class XI Chapter 13 – Limits and Derivatives Maths ______________________________________________________________________________ ? ? ? ? ? ? ? ? 2 2 2 cos .cos sin sin ' cos cos cos cos cos x x a x x a fx x x a x x a x ? ? ? ? ?? ? ? Question 22: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): x 4 (5 sin x - 3 cos x) Solution 22: Let 4 ( ) (5sin 3cos ) f x x x x ?? By product rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? 44 44 43 3 '( ) (5sin 3cos ) (5sin 3cos ) 5 sin 3 cos (5sin 3cos ) 5cos 3 sin (5sin 3cos ) 4 5 cos 3 sin 20sin 12cos dd f x x x x x x x dx dx d d d x x x x x x dx dx dx x x x x x x x x x x x x x ? ? ? ? ?? ? ? ? ? ?? ?? ? ? ? ? ? ?? ?? ? ? ? ? Question 23: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (x 2 + 1) cos x Solution 23: Let f(x) = ? ? 2 1 cos xx ? By product rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? 22 2 2 ' 1 cos cos 1 1 sin cos 2 sin sin 2 cos dd f x x x x x dx dx x x x x x x x x x ? ? ? ? ? ? ? ? ? ? ? ? Class XI Chapter 13 – Limits and Derivatives Maths ______________________________________________________________________________ Question 24: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (ax 2 + sin x) (p + q cos x) Solution 24: Let ? ? ? ? 2 sin f x ax x ?? (p + q cos x) By product rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 22 2 2 ' sin cos cos sin sin sin cos 2 cos sin sin cos 2 cos dd f x ax x p q x p q x ax x dx dx ax x q x p q x ax x q x ax x p q x ax x ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Question 25: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (x + cos x) (x – tan x) Solution 25: Let f(x) = (x + cos x) (x – tan x) By product rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ' (x + cos x) tan tan cos x + cos x tan tan 1 sin x + cos x 1 tan tan 1 sin ... Let g x tan . Accordingly, g(x+h)=tan x+h dd f x x x x x x x dx dx dd x x x x x dx dx d x x x x i dx x ? ? ? ? ? ?? ? ? ? ? ? ?? ?? ?? ? ? ? ? ? ?? ?? ? By first principle, ? ? ? ? ? ? ? ? ? ? 0 0 0 ' lim tan tan lim 1 sin( ) sin lim cos cos h h h g x h g x gx h x h x h x h x h x h x ? ? ? ?? ? ?? ? ?? ? ?? ?? ? ?? Class XI Chapter 13 – Limits and Derivatives Maths ______________________________________________________________________________ ? ? ? ? 0 sin( )cos sin cos 1 lim cos cos h x h x x x h h x h x ? ?? ? ? ? ? ?? ? ?? ? ? ? ? ? ? ? ? ? ? 0 0 00 2 2 1 1 in( ) lim cos cos 1 1 sinh lim cos cos 1 sinh 1 . lim . lim cos cos 11 .1. cos cos 0 1 cos sec h h hh s x h x x h x h x h x h x h x h xx x x ii ? ? ?? ?? ?? ? ?? ? ?? ?? ? ?? ? ?? ?? ?? ? ?? ?? ?? ? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 2 2 2 Therefore, from (i) and ii ,We obtain f' x cos (1 sec ) tan 1 sin cos tan tan 1 sin tan cos tan 1 sin x x x x x x x x x x x x x x x x x x ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Question 26: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non zero constants and m and n are integers): 4 5sin 3 7cos xx xx ? ? Solution 26: Let f(x) = 4 5sin 3 7cos xx xx ? ? By quotient rule, ? ? ? ? ? ? ? ? ? ? ? ? 2 3 7cos 4 5sin 4 5sin 3 7cos ' 3 7cos dd x x x x x x x x dx dx fx xx ? ? ? ? ? ? ? Class XI Chapter 13 – Limits and Derivatives Maths ______________________________________________________________________________ ? ? ? ? ? ? ? ? ? ? 2 3 7cos 4 5 sin 4 5sin 3 7 cos 3 7cos d d d d x x x x x x x x dx dx dx dx xx ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 2 2 2 2 2 22 2 3 7cos 4 5cos 4 5sin 3 7sin 3 7cos 12 15 cos 28cos 35cos 12 28 sin 15sin 35 cos sin 3 7cos 15 cos 28cos 28 sin 15sin 35(cos sin ) 3 7cos 35 15 cos 28cos 28 sin 15sin 3 x x x x x x x xx x x x x x x x x x x x xx x x x x x x x x xx x x x x x x ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 2 7cos xx ? Question 27: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed nonzero constants and m and n are integers): 2 cos 4 sin x x ? ?? ?? ?? Solution 27: Let f(x) = 2 cos 4 sin x x ? ?? ?? ?? By quotient rule, ? ? ? ? ? ? ? ? 22 2 2 2 2 sin sin ' cos . 4 sin sin .2 cos cos . 4 sin cos 2sin cos 4 sin dd x x x x dx dx fx x x x x x x x x x x x ? ? ? ?? ? ?? ?? ? ?? ?? ?? ?? ?? ?? ? ?? ? ???? ?? ?? ? ?Read More
|
75 videos|238 docs|91 tests
|
FAQs on NCERT Solutions Class 11 Maths Chapter 12 - Limits and Derivatives
| 1. What is the concept of limits in calculus? |  |
Ans. In calculus, the concept of limits refers to the value that a function approaches as the input variable gets closer and closer to a certain value. It helps us understand the behavior of functions and their values at specific points.
| 2. How do limits help in finding derivatives? | |
Ans. Limits play a crucial role in finding derivatives. The derivative of a function at a point is defined as the limit of the slope of a secant line passing through that point and a nearby point on the function. By taking the limit as the two points get closer, we can find the instantaneous rate of change, which is the derivative.
| 3. What is the difference between a one-sided limit and a two-sided limit? | |
Ans. A one-sided limit is concerned with the behavior of a function as the input approaches a certain value from either the left or the right side. It only considers the values of the function from one side of the given point. On the other hand, a two-sided limit considers the values of the function from both the left and right sides of the given point.
| 4. Can limits be used to determine the continuity of a function? | |
Ans. Yes, limits can be used to determine the continuity of a function. If the limit of a function at a certain point exists and is equal to the value of the function at that point, then the function is continuous at that point. Limits help us analyze the behavior of a function and identify any potential discontinuities.
| 5. How are limits useful in real-life applications? | |
Ans. Limits have various real-life applications. In physics, limits are used to calculate instantaneous velocity and acceleration. In economics, limits help in determining marginal cost and marginal revenue. In engineering, limits are used to analyze the stability and efficiency of systems. Overall, limits provide a mathematical framework to understand and solve problems in various fields.
Related Exams
About this Document
|
3.3K Views |
|
4.98/5 Rating |
|
Dec 27, 2024 Last updated |
Document Description: NCERT Solutions Miscellaneous Exercise 2: Limits and Derivatives for Commerce 2024 is part of Mathematics (Maths) Class 11 preparation.
The notes and questions for NCERT Solutions Miscellaneous Exercise 2: Limits and Derivatives have been prepared according to the Commerce exam syllabus. Information about NCERT Solutions Miscellaneous Exercise 2: Limits and Derivatives covers topics
like and NCERT Solutions Miscellaneous Exercise 2: Limits and Derivatives Example, for Commerce 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for NCERT Solutions Miscellaneous Exercise 2: Limits and Derivatives.
Introduction of NCERT Solutions Miscellaneous Exercise 2: Limits and Derivatives in English is available as part of our Mathematics (Maths) Class 11
for Commerce & NCERT Solutions Miscellaneous Exercise 2: Limits and Derivatives in Hindi for Mathematics (Maths) Class 11 course.
Download more important topics related with notes, lectures and mock test series for Commerce
Exam by signing up for free. Commerce: NCERT Solutions Class 11 Maths Chapter 12 - Limits and Derivatives
Description
Full syllabus notes, lecture & questions for NCERT Solutions Class 11 Maths Chapter 12 - Limits and Derivatives - Commerce | Plus excerises question with solution to help you revise complete syllabus for Mathematics (Maths) Class 11 | Best notes, free PDF download
Information about NCERT Solutions Miscellaneous Exercise 2: Limits and Derivatives
In this doc you can find the meaning of NCERT Solutions Miscellaneous Exercise 2: Limits and Derivatives defined & explained in the simplest way possible. Besides explaining types of
NCERT Solutions Miscellaneous Exercise 2: Limits and Derivatives theory, EduRev gives you an ample number of questions to practice NCERT Solutions Miscellaneous Exercise 2: Limits and Derivatives tests, examples and also practice Commerce
tests
|
Explore Courses for Commerce exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
Related Searches
Previous Year Questions with Solutions
,Objective type Questions
,Extra Questions
,ppt
,shortcuts and tricks
,Summary
,practice quizzes
,Viva Questions
,MCQs
,Exam
,Semester Notes
,study material
,NCERT Solutions Class 11 Maths Chapter 12 - Limits and Derivatives
,Important questions
,Free
,NCERT Solutions Class 11 Maths Chapter 12 - Limits and Derivatives
,past year papers
,video lectures
,NCERT Solutions Class 11 Maths Chapter 12 - Limits and Derivatives
,mock tests for examination
,Sample Paper
;
Additional Information about NCERT Solutions Miscellaneous Exercise 2: Limits and Derivatives for Commerce Preparation
NCERT Solutions Miscellaneous Exercise 2: Limits and Derivatives Free PDF Download
The NCERT Solutions Miscellaneous Exercise 2: Limits and Derivatives is an invaluable resource that delves deep into the core of the Commerce exam.
These study notes are curated by experts and cover all the essential topics and concepts, making your preparation more efficient and effective.
With the help of these notes, you can grasp complex subjects quickly, revise important points easily,
and reinforce your understanding of key concepts. The study notes are presented in a concise and easy-to-understand manner,
allowing you to optimize your learning process. Whether you're looking for best-recommended books, sample papers, study material,
or toppers' notes, this PDF has got you covered. Download the NCERT Solutions Miscellaneous Exercise 2: Limits and Derivatives now and kickstart your journey towards success in the Commerce exam.
Importance of NCERT Solutions Miscellaneous Exercise 2: Limits and Derivatives
The importance of NCERT Solutions Miscellaneous Exercise 2: Limits and Derivatives cannot be overstated, especially for Commerce aspirants.
This document holds the key to success in the Commerce exam.
It offers a detailed understanding of the concept, providing invaluable insights into the topic.
By knowing the concepts well in advance, students can plan their preparation effectively.
Utilize this indispensable guide for a well-rounded preparation and achieve your desired results.
NCERT Solutions Miscellaneous Exercise 2: Limits and Derivatives Notes
NCERT Solutions Miscellaneous Exercise 2: Limits and Derivatives Notes offer in-depth insights into the specific topic to help you master it with ease.
This comprehensive document covers all aspects related to NCERT Solutions Miscellaneous Exercise 2: Limits and Derivatives.
It includes detailed information about the exam syllabus, recommended books, and study materials for a well-rounded preparation.
Practice papers and question papers enable you to assess your progress effectively.
Additionally, the paper analysis provides valuable tips for tackling the exam strategically.
Access to Toppers' notes gives you an edge in understanding complex concepts.
Whether you're a beginner or aiming for advanced proficiency, NCERT Solutions Miscellaneous Exercise 2: Limits and Derivatives Notes on EduRev are your ultimate resource for success.
NCERT Solutions Miscellaneous Exercise 2: Limits and Derivatives Commerce Questions
The "NCERT Solutions Miscellaneous Exercise 2: Limits and Derivatives Commerce Questions" guide is a valuable resource for all aspiring students preparing for the
Commerce exam. It focuses on providing a wide range of practice questions to help students gauge
their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation.
The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level.
Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.
Study NCERT Solutions Miscellaneous Exercise 2: Limits and Derivatives on the App
Students of Commerce can study NCERT Solutions Miscellaneous Exercise 2: Limits and Derivatives alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the NCERT Solutions Miscellaneous Exercise 2: Limits and Derivatives,
students can also utilize the EduRev App for other study materials such as previous year question papers, syllabus, important questions, etc.
The EduRev App will make your learning easier as you can access it from anywhere you want.
The content of NCERT Solutions Miscellaneous Exercise 2: Limits and Derivatives is prepared as per the latest Commerce syllabus.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup