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NCERT Solutions Class 11 Maths Chapter 12 - Limits and Derivatives
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Page 1 Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ Question 8: Find the derivative of nn xa xa ? ? for some constant a. Solution 8: Let f(x) = nn xa xa ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 2 1 2 1 2 '( ) By quotient rule, ' 0 nn n n n n n n n n n n n d x a fx dx x a dd x a x a x a x a dx dx fx xa x a nx x a xa nx anx x a xa ? ? ?? ? ?? ?? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Question 9: Find the derivative of (i) 3 2 4 x ? (ii) (5x 3 + 3x – 1) (x – 1) (iii) â x €’’3 (5 + 3x) (iv) x5 (3 –6 a x €’’9 ) (v) x–4 (3 – 4x–5) (vi) 2 2 1 3 1 x xx ? ?? Solution 9: (i) Let f(x) = 3 2 4 x ? 3 '( ) 2 4 d f x x dx ?? ?? ?? ?? = ? ? 3 2 4 dd x dx dx ?? ? ?? ?? =2-0 =2 (ii) Let f (x) = (5x 3 + 3x – 1) (x – 1) Page 2 Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ Question 8: Find the derivative of nn xa xa ? ? for some constant a. Solution 8: Let f(x) = nn xa xa ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 2 1 2 1 2 '( ) By quotient rule, ' 0 nn n n n n n n n n n n n d x a fx dx x a dd x a x a x a x a dx dx fx xa x a nx x a xa nx anx x a xa ? ? ?? ? ?? ?? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Question 9: Find the derivative of (i) 3 2 4 x ? (ii) (5x 3 + 3x – 1) (x – 1) (iii) â x €’’3 (5 + 3x) (iv) x5 (3 –6 a x €’’9 ) (v) x–4 (3 – 4x–5) (vi) 2 2 1 3 1 x xx ? ?? Solution 9: (i) Let f(x) = 3 2 4 x ? 3 '( ) 2 4 d f x x dx ?? ?? ?? ?? = ? ? 3 2 4 dd x dx dx ?? ? ?? ?? =2-0 =2 (ii) Let f (x) = (5x 3 + 3x – 1) (x – 1) Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ By Leibnitz product rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 33 32 32 3 3 2 32 '( ) 5x + 3x-1 1 1 5x + 3x-1 5x + 3x-1 1 1 5.3x + 3-0 5x + 3x-1 1 15 3 5x + 3x-1+15x 3 15 3 20 15 6 4 dd f x x x dx dx x xx xx x x x ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? (iii) Let f (x) = â x €’’3 (5 + 3x) By Leibnitz product rule, ? ? ? ? ? ? -3 -3 f'(x)=x 5 + 3x 5 3 x dd x dx dx ?? ? ? ? ? ? ? ? ? ? ? ? ? -3 -3-1 -3 -4 -3 -4 -3 -3 -4 =x 0 + 3 5 3 3x x 3 5 3 3x 3x 15x 9x 6x 15x x x ?? ? ? ? ? ? ? ? ? ? ? ? ? ? -3 -3 4 5 3x 2 3x 25 3 52 x x x x x ?? ? ? ? ?? ?? ? ?? ? ?? (iv) Let f (x) = x 5 (3 – 6 a x €’’9 ) By Leibnitz product rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 5 -9 9 5 5 -9-1 -9 4 5 -10 4 -5 -5 4 -5 -5 4 -5 5 '( ) x 3 -6x 3 6 x x 0 6 9 x 3 6x 5x x 54x 15x 30x 54x 15x 30x 24x 15x 24 24x x dd f x x dx dx ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ?? (v) Let f (x) = a x €’’4 (3 – 4 a x €’’5 ) By Leibnitz product rule, Page 3 Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ Question 8: Find the derivative of nn xa xa ? ? for some constant a. Solution 8: Let f(x) = nn xa xa ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 2 1 2 1 2 '( ) By quotient rule, ' 0 nn n n n n n n n n n n n d x a fx dx x a dd x a x a x a x a dx dx fx xa x a nx x a xa nx anx x a xa ? ? ?? ? ?? ?? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Question 9: Find the derivative of (i) 3 2 4 x ? (ii) (5x 3 + 3x – 1) (x – 1) (iii) â x €’’3 (5 + 3x) (iv) x5 (3 –6 a x €’’9 ) (v) x–4 (3 – 4x–5) (vi) 2 2 1 3 1 x xx ? ?? Solution 9: (i) Let f(x) = 3 2 4 x ? 3 '( ) 2 4 d f x x dx ?? ?? ?? ?? = ? ? 3 2 4 dd x dx dx ?? ? ?? ?? =2-0 =2 (ii) Let f (x) = (5x 3 + 3x – 1) (x – 1) Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ By Leibnitz product rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 33 32 32 3 3 2 32 '( ) 5x + 3x-1 1 1 5x + 3x-1 5x + 3x-1 1 1 5.3x + 3-0 5x + 3x-1 1 15 3 5x + 3x-1+15x 3 15 3 20 15 6 4 dd f x x x dx dx x xx xx x x x ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? (iii) Let f (x) = â x €’’3 (5 + 3x) By Leibnitz product rule, ? ? ? ? ? ? -3 -3 f'(x)=x 5 + 3x 5 3 x dd x dx dx ?? ? ? ? ? ? ? ? ? ? ? ? ? -3 -3-1 -3 -4 -3 -4 -3 -3 -4 =x 0 + 3 5 3 3x x 3 5 3 3x 3x 15x 9x 6x 15x x x ?? ? ? ? ? ? ? ? ? ? ? ? ? ? -3 -3 4 5 3x 2 3x 25 3 52 x x x x x ?? ? ? ? ?? ?? ? ?? ? ?? (iv) Let f (x) = x 5 (3 – 6 a x €’’9 ) By Leibnitz product rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 5 -9 9 5 5 -9-1 -9 4 5 -10 4 -5 -5 4 -5 -5 4 -5 5 '( ) x 3 -6x 3 6 x x 0 6 9 x 3 6x 5x x 54x 15x 30x 54x 15x 30x 24x 15x 24 24x x dd f x x dx dx ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ?? (v) Let f (x) = a x €’’4 (3 – 4 a x €’’5 ) By Leibnitz product rule, Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? -4 -5 5 -4 -4 -5-1 -5 4 1 -4 -6 -5 -5 -10 -5 -10 -10 -5 -5 10 '( ) x 3 -4x 3 4 x x 0 4 5 x 3 4x 4 x 20x 3 4x 4x 20x 12x 16x 36x 12x 12 36 xx dd f x x dx dx x ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? (vi) Let f (x) = 2 2 1 3 1 x xx ? ?? f’(x) = d dx 2 2 1 3 1 dx x dx x ?? ?? ? ?? ?? ?? ?? ?? By quotient rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 22 22 2 22 22 22 22 22 22 1 2 2 1 3 1 3 1 ' 1 3 1 3 1 2 3 1 0 2 1 1 3 1 2 6 2 3 1 3 1 2 3 2 1 3 1 32 2 1 3 1 d d d d x x x x x x dx dx dx dx fx xx x x x x xx x x x xx xx xx xx xx ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ? ? ? ? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ???? ?? ? ? ? ???? ?? ?? ?? ?? ?? ???? ?? ?? ?? ? ? ?? ?? Question 10: Find the derivative of cos x from first principle. Solution 10: Let f (x) = cos x. Accordingly, from the first principle, ? ? ? ? ? ? 0 ' lim h f x h f x fx h ? ?? ? Page 4 Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ Question 8: Find the derivative of nn xa xa ? ? for some constant a. Solution 8: Let f(x) = nn xa xa ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 2 1 2 1 2 '( ) By quotient rule, ' 0 nn n n n n n n n n n n n d x a fx dx x a dd x a x a x a x a dx dx fx xa x a nx x a xa nx anx x a xa ? ? ?? ? ?? ?? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Question 9: Find the derivative of (i) 3 2 4 x ? (ii) (5x 3 + 3x – 1) (x – 1) (iii) â x €’’3 (5 + 3x) (iv) x5 (3 –6 a x €’’9 ) (v) x–4 (3 – 4x–5) (vi) 2 2 1 3 1 x xx ? ?? Solution 9: (i) Let f(x) = 3 2 4 x ? 3 '( ) 2 4 d f x x dx ?? ?? ?? ?? = ? ? 3 2 4 dd x dx dx ?? ? ?? ?? =2-0 =2 (ii) Let f (x) = (5x 3 + 3x – 1) (x – 1) Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ By Leibnitz product rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 33 32 32 3 3 2 32 '( ) 5x + 3x-1 1 1 5x + 3x-1 5x + 3x-1 1 1 5.3x + 3-0 5x + 3x-1 1 15 3 5x + 3x-1+15x 3 15 3 20 15 6 4 dd f x x x dx dx x xx xx x x x ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? (iii) Let f (x) = â x €’’3 (5 + 3x) By Leibnitz product rule, ? ? ? ? ? ? -3 -3 f'(x)=x 5 + 3x 5 3 x dd x dx dx ?? ? ? ? ? ? ? ? ? ? ? ? ? -3 -3-1 -3 -4 -3 -4 -3 -3 -4 =x 0 + 3 5 3 3x x 3 5 3 3x 3x 15x 9x 6x 15x x x ?? ? ? ? ? ? ? ? ? ? ? ? ? ? -3 -3 4 5 3x 2 3x 25 3 52 x x x x x ?? ? ? ? ?? ?? ? ?? ? ?? (iv) Let f (x) = x 5 (3 – 6 a x €’’9 ) By Leibnitz product rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 5 -9 9 5 5 -9-1 -9 4 5 -10 4 -5 -5 4 -5 -5 4 -5 5 '( ) x 3 -6x 3 6 x x 0 6 9 x 3 6x 5x x 54x 15x 30x 54x 15x 30x 24x 15x 24 24x x dd f x x dx dx ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ?? (v) Let f (x) = a x €’’4 (3 – 4 a x €’’5 ) By Leibnitz product rule, Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? -4 -5 5 -4 -4 -5-1 -5 4 1 -4 -6 -5 -5 -10 -5 -10 -10 -5 -5 10 '( ) x 3 -4x 3 4 x x 0 4 5 x 3 4x 4 x 20x 3 4x 4x 20x 12x 16x 36x 12x 12 36 xx dd f x x dx dx x ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? (vi) Let f (x) = 2 2 1 3 1 x xx ? ?? f’(x) = d dx 2 2 1 3 1 dx x dx x ?? ?? ? ?? ?? ?? ?? ?? By quotient rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 22 22 2 22 22 22 22 22 22 1 2 2 1 3 1 3 1 ' 1 3 1 3 1 2 3 1 0 2 1 1 3 1 2 6 2 3 1 3 1 2 3 2 1 3 1 32 2 1 3 1 d d d d x x x x x x dx dx dx dx fx xx x x x x xx x x x xx xx xx xx xx ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ? ? ? ? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ???? ?? ? ? ? ???? ?? ?? ?? ?? ?? ???? ?? ?? ?? ? ? ?? ?? Question 10: Find the derivative of cos x from first principle. Solution 10: Let f (x) = cos x. Accordingly, from the first principle, ? ? ? ? ? ? 0 ' lim h f x h f x fx h ? ?? ? Chapter 12 – Limits and Derivatives Maths ______________________________________________________________________________ ? ? ? ? ? ? ? ? ? ? 0 0 0 0 00 0 cos cos lim cos cosh sin sinh cos lim cos 1 cosh sin sinh lim cos (1 cosh sin sinh lim 1 cosh sinh cos lim sin lim 1 cosh cos 0 sin 1 lim 0 h h h h hh h x h x h x x x h xx h xx hh xx hh xx h ? ? ? ? ?? ? ?? ? ?? ?? ? ?? ?? ? ? ? ?? ? ?? ?? ???? ?? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 0 sinh lim 1 sin '( ) sin h and h x f x x ? ?? ? ?? ?? ?? ? ? ? Question 11: Find the derivative of the following functions: (i) sin x cos x (ii) sec x (iii) 5 sec x + 4 cos x (iv) cosec x (v) 3cot x + 5cosec x (vi) 5sin x - 6cos x + 7 (vii) 2tan x - 7sec x Solution 11: (i) Let f (x) = sin x cos x. Accordingly, from the first principle, ? ? ? ? ? ? ? ? ? ? ? ? 0 0 0 ' lim sin cos sin cos lim 1 lim 2sin( )cos 2sin cos 2 h h h f x h f x fx h x h x h x x h x h x h x x h ? ? ? ?? ? ? ? ? ? ? ? ? ? ?? ?? ? ? 0 0 1 lim sin2 sin2 2 1 2 2 2 2 2h 2 lim 2cos .sin 2 2 2 h h x h x h x h x x x h ? ? ? ? ? ?? ?? ? ? ? ? ?? ? ?? ?? 0 1 4 2 2h lim cos .sin 2 2 2 h xh h ? ??? ? ?? ?? Page 5 Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ Question 8: Find the derivative of nn xa xa ? ? for some constant a. Solution 8: Let f(x) = nn xa xa ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 2 1 2 1 2 '( ) By quotient rule, ' 0 nn n n n n n n n n n n n d x a fx dx x a dd x a x a x a x a dx dx fx xa x a nx x a xa nx anx x a xa ? ? ?? ? ?? ?? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Question 9: Find the derivative of (i) 3 2 4 x ? (ii) (5x 3 + 3x – 1) (x – 1) (iii) â x €’’3 (5 + 3x) (iv) x5 (3 –6 a x €’’9 ) (v) x–4 (3 – 4x–5) (vi) 2 2 1 3 1 x xx ? ?? Solution 9: (i) Let f(x) = 3 2 4 x ? 3 '( ) 2 4 d f x x dx ?? ?? ?? ?? = ? ? 3 2 4 dd x dx dx ?? ? ?? ?? =2-0 =2 (ii) Let f (x) = (5x 3 + 3x – 1) (x – 1) Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ By Leibnitz product rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 33 32 32 3 3 2 32 '( ) 5x + 3x-1 1 1 5x + 3x-1 5x + 3x-1 1 1 5.3x + 3-0 5x + 3x-1 1 15 3 5x + 3x-1+15x 3 15 3 20 15 6 4 dd f x x x dx dx x xx xx x x x ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? (iii) Let f (x) = â x €’’3 (5 + 3x) By Leibnitz product rule, ? ? ? ? ? ? -3 -3 f'(x)=x 5 + 3x 5 3 x dd x dx dx ?? ? ? ? ? ? ? ? ? ? ? ? ? -3 -3-1 -3 -4 -3 -4 -3 -3 -4 =x 0 + 3 5 3 3x x 3 5 3 3x 3x 15x 9x 6x 15x x x ?? ? ? ? ? ? ? ? ? ? ? ? ? ? -3 -3 4 5 3x 2 3x 25 3 52 x x x x x ?? ? ? ? ?? ?? ? ?? ? ?? (iv) Let f (x) = x 5 (3 – 6 a x €’’9 ) By Leibnitz product rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 5 -9 9 5 5 -9-1 -9 4 5 -10 4 -5 -5 4 -5 -5 4 -5 5 '( ) x 3 -6x 3 6 x x 0 6 9 x 3 6x 5x x 54x 15x 30x 54x 15x 30x 24x 15x 24 24x x dd f x x dx dx ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ?? (v) Let f (x) = a x €’’4 (3 – 4 a x €’’5 ) By Leibnitz product rule, Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? -4 -5 5 -4 -4 -5-1 -5 4 1 -4 -6 -5 -5 -10 -5 -10 -10 -5 -5 10 '( ) x 3 -4x 3 4 x x 0 4 5 x 3 4x 4 x 20x 3 4x 4x 20x 12x 16x 36x 12x 12 36 xx dd f x x dx dx x ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? (vi) Let f (x) = 2 2 1 3 1 x xx ? ?? f’(x) = d dx 2 2 1 3 1 dx x dx x ?? ?? ? ?? ?? ?? ?? ?? By quotient rule, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 22 22 2 22 22 22 22 22 22 1 2 2 1 3 1 3 1 ' 1 3 1 3 1 2 3 1 0 2 1 1 3 1 2 6 2 3 1 3 1 2 3 2 1 3 1 32 2 1 3 1 d d d d x x x x x x dx dx dx dx fx xx x x x x xx x x x xx xx xx xx xx ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ? ? ? ? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ???? ?? ? ? ? ???? ?? ?? ?? ?? ?? ???? ?? ?? ?? ? ? ?? ?? Question 10: Find the derivative of cos x from first principle. Solution 10: Let f (x) = cos x. Accordingly, from the first principle, ? ? ? ? ? ? 0 ' lim h f x h f x fx h ? ?? ? Chapter 12 – Limits and Derivatives Maths ______________________________________________________________________________ ? ? ? ? ? ? ? ? ? ? 0 0 0 0 00 0 cos cos lim cos cosh sin sinh cos lim cos 1 cosh sin sinh lim cos (1 cosh sin sinh lim 1 cosh sinh cos lim sin lim 1 cosh cos 0 sin 1 lim 0 h h h h hh h x h x h x x x h xx h xx hh xx hh xx h ? ? ? ? ?? ? ?? ? ?? ?? ? ?? ?? ? ? ? ?? ? ?? ?? ???? ?? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 0 sinh lim 1 sin '( ) sin h and h x f x x ? ?? ? ?? ?? ?? ? ? ? Question 11: Find the derivative of the following functions: (i) sin x cos x (ii) sec x (iii) 5 sec x + 4 cos x (iv) cosec x (v) 3cot x + 5cosec x (vi) 5sin x - 6cos x + 7 (vii) 2tan x - 7sec x Solution 11: (i) Let f (x) = sin x cos x. Accordingly, from the first principle, ? ? ? ? ? ? ? ? ? ? ? ? 0 0 0 ' lim sin cos sin cos lim 1 lim 2sin( )cos 2sin cos 2 h h h f x h f x fx h x h x h x x h x h x h x x h ? ? ? ?? ? ? ? ? ? ? ? ? ? ?? ?? ? ? 0 0 1 lim sin2 sin2 2 1 2 2 2 2 2h 2 lim 2cos .sin 2 2 2 h h x h x h x h x x x h ? ? ? ? ? ?? ?? ? ? ? ? ?? ? ?? ?? 0 1 4 2 2h lim cos .sin 2 2 2 h xh h ? ??? ? ?? ?? Chapter 2 1 – Limits and Derivatives Maths ______________________________________________________________________________ ? ? ? ? ? ? 0 00 1 lim cos 2 sinh 2 sinh limcos 2 .lim cos 2 0 .1 cos2 h hh xh h xh h x x ? ?? ?? ?? ?? ?? ?? ? (ii) Let f (x) = sec x. Accordingly, from the first principle, ? ? ? ? ? ? ? ? ? ? ? ? 0 0 0 0 0 0 ' lim sec sec lim 1 1 1 lim cos( ) cos cos cos 1 lim cos cos 2sin sin 11 22 .lim cos cos( ) 2 2sin 11 2 .lim cos h h h h h h f x h f x fx h x h x h h x h x x x h h x x h x x h x x h x h x h xh xh ? ? ? ? ? ? ?? ? ?? ? ?? ?? ?? ? ?? ?? ?? ? ?? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? ? ? ?? ?? ?? ? ?? ? ? ?? ? 0 00 sin 2 cos( ) sin 2 2 sin 2 2 11 .lim cos 2 cos( ) 2 sin sin 1 22 .lim .lim cos cos( ) 2 1 sin .1. cos cos sec tan h hh h xh h xh h x h x h h x h h x x h x xx xx ? ?? ?? ?? ? ? ? ? ?? ?? ?? ? ?? ?? ?? ?? ?? ? ?? ?? ? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ?? ?? ? ? (iii) Let f (x) = 5 sec x + 4 cos x. Accordingly, from the first principle,Read More
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FAQs on NCERT Solutions Class 11 Maths Chapter 12 - Limits and Derivatives
| 1. What is the basic definition of a limit in calculus? |  |
Ans. The basic definition of a limit in calculus is the value that a function approaches as the input value approaches a certain point or infinity.
| 2. How can limits be used to find the derivative of a function? | |
Ans. Limits are used in calculus to find the derivative of a function by taking the limit of the difference quotient as the interval approaches zero.
| 3. What is the significance of derivatives in calculus? | |
Ans. Derivatives in calculus represent the rate of change of a function at a given point, providing information about the slope of the function and its behavior.
| 4. How can one determine if a function is continuous at a certain point using limits and derivatives? | |
Ans. A function is continuous at a certain point if the limit of the function as it approaches that point is equal to the function value at that point, and the derivative exists at that point.
| 5. Can limits and derivatives be used to solve real-world problems? | |
Ans. Yes, limits and derivatives are frequently used in solving real-world problems involving rates of change, optimization, and motion. These concepts are essential in various fields such as physics, engineering, and economics.
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