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NCERT Solutions Class 11 Maths Chapter 12 - Limits and Derivatives
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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
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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.
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