Tamil Nadu Board of Secondary EducationHSC Science Class 11th
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Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide chapter 10 - Differential Calculus - Differentiability and Methods of Differentiation [Latest edition]

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Class 11th Mathematics Volume 1 and 2 Answers Guide - Shaalaa.com
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Chapter 10: Differential Calculus - Differentiability and Methods of Differentiation

Exercise 10.1Exercise 10.2Exercise 10.3Exercise 10.4Exercise 10.5
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Exercise 10.1 [Page 147]

Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 10 Differential Calculus - Differentiability and Methods of DifferentiationExercise 10.1 [Page 147]

Exercise 10.1 | Q 1. (i) | Page 147

Find the derivatives of the following functions using first principle.

f(x) = 6

Exercise 10.1 | Q 1. (ii) | Page 147

Find the derivatives of the following functions using first principle.

f(x) = – 4x + 7

Exercise 10.1 | Q 1. (iii) | Page 147

Find the derivatives of the following functions using first principle.

f(x) = – x2 + 2

Exercise 10.1 | Q 2. (i) | Page 147

Find the derivatives from the left and from the right at x = 1 (if they exist) of the following functions. Are the functions differentiable at x = 1?

`f(x) = |x - 1|`

Exercise 10.1 | Q 2. (ii) | Page 147

Find the derivatives from the left and from the right at x = 1 (if they exist) of the following functions. Are the functions differentiable at x = 1?

`f(x) = sqrt(1 - x^2)`

Exercise 10.1 | Q 2. (iii) | Page 147

Find the derivatives from the left and from the right at x = 1 (if they exist) of the following functions. Are the functions differentiable at x = 1?

`f(x) = {{:(x",", x ≤ 1),(x^2",", x > 1):}`

Exercise 10.1 | Q 3. (i) | Page 147

Determine whether the following function is differentiable at the indicated values.

f(x) = x |x| at x = 0

Exercise 10.1 | Q 3. (ii) | Page 147

Determine whether the following function is differentiable at the indicated values.

f(x) = |x2 – 1| at x = 1

Exercise 10.1 | Q 3. (iii) | Page 147

Determine whether the following function is differentiable at the indicated values.

f(x) = |x| + |x – 1| at x = 0, 1

Exercise 10.1 | Q 3. (iv) | Page 147

Determine whether the following function is differentiable at the indicated values.

f(x) = sin |x| at x = 0

Exercise 10.1 | Q 4. (i) | Page 147

Show that the following functions are not differentiable at the indicated value of x.

`f(x) = {{:(-x + 2, x ≤ 2),(2x - 4, x > 2):}` , x = 2

Exercise 10.1 | Q 4. (ii) | Page 147

Show that the following functions are not differentiable at the indicated value of x.

`f(x) = {{:(3x",", x < 0),(-4x",", x ≥ 0):}` , x = 0

Exercise 10.1 | Q 5 | Page 147

The graph of f is shown below. State with reasons that x values (the numbers), at which f is not differentiable.

Exercise 10.1 | Q 6 | Page 147

If f(x) = |x + 100| + x2, test whether f’(–100) exists.

Exercise 10.1 | Q 7. (i) | Page 147

Examine the differentiability of functions in R by drawing the diagram

|sin x|

Exercise 10.1 | Q 7. (ii) | Page 147

Examine the differentiability of functions in R by drawing the diagram

|cos x|

Exercise 10.2 [Page 160]

Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 10 Differential Calculus - Differentiability and Methods of DifferentiationExercise 10.2 [Page 160]

Exercise 10.2 | Q 1 | Page 160

Find the derivatives of the following functions with respect to corresponding independent variables:

f(x) = x – 3 sin x

Exercise 10.2 | Q 2 | Page 160

Find the derivatives of the following functions with respect to corresponding independent variables:

y = sin x + cos x

Exercise 10.2 | Q 3 | Page 160

Find the derivatives of the following functions with respect to corresponding independent variables:

f(x) = x sin x

Exercise 10.2 | Q 4 | Page 160

Find the derivatives of the following functions with respect to corresponding independent variables:

y = cos x – 2 tan x

Exercise 10.2 | Q 5 | Page 160

Find the derivatives of the following functions with respect to corresponding independent variables:

g(t) = t3 cos t

Exercise 10.2 | Q 6 | Page 160

Find the derivatives of the following functions with respect to corresponding independent variables:

g(t) = 4 sec t + tan t

Exercise 10.2 | Q 7 | Page 160

Find the derivatives of the following functions with respect to corresponding independent variables:

y = ex sin x

Exercise 10.2 | Q 8 | Page 160

Find the derivatives of the following functions with respect to corresponding independent variables:

y = `tan x/x`

Exercise 10.2 | Q 9 | Page 160

Find the derivatives of the following functions with respect to corresponding independent variables:

y = `sinx/(1 + cosx)`

Exercise 10.2 | Q 10 | Page 160

Find the derivatives of the following functions with respect to corresponding independent variables:

y = `x/(sin x + cosx)`

Exercise 10.2 | Q 11 | Page 160

Find the derivatives of the following functions with respect to corresponding independent variables:

y = `(tanx - 1)/secx`

Exercise 10.2 | Q 12 | Page 160

Find the derivatives of the following functions with respect to corresponding independent variables:

y = `sinx/x^2`

Exercise 10.2 | Q 13 | Page 160

Find the derivatives of the following functions with respect to corresponding independent variables:

y = tan θ (sin θ + cos θ)

Exercise 10.2 | Q 14 | Page 160

Find the derivatives of the following functions with respect to corresponding independent variables:

y = cosec x . cot x

Exercise 10.2 | Q 15 | Page 160

Find the derivatives of the following functions with respect to corresponding independent variables:

y = x sin x cos x

Exercise 10.2 | Q 16 | Page 160

Find the derivatives of the following functions with respect to corresponding independent variables:

y = e-x . log x

Exercise 10.2 | Q 17 | Page 160

Find the derivatives of the following functions with respect to corresponding independent variables:

y = (x2 + 5) log(1 + x) e–3x 

Exercise 10.2 | Q 18 | Page 160

Find the derivatives of the following functions with respect to corresponding independent variables:

y = sin x

Exercise 10.2 | Q 19 | Page 160

Find the derivatives of the following functions with respect to corresponding independent variables:

y = log10 x

Exercise 10.2 | Q 20 | Page 160

Find the derivatives of the following functions with respect to corresponding independent variables:

Draw the function f'(x) if f(x) = 2x2 – 5x + 3

Exercise 10.3 [Pages 163 - 164]

Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 10 Differential Calculus - Differentiability and Methods of DifferentiationExercise 10.3 [Pages 163 - 164]

Exercise 10.3 | Q 1 | Page 163

Differentiate the following:
y = (x2 + 4x + 6)5

Exercise 10.3 | Q 2 | Page 163

Differentiate the following:
y = tan 3x

Exercise 10.3 | Q 3 | Page 163

Differentiate the following:
y = cos (tan x)

Exercise 10.3 | Q 4 | Page 163

Differentiate the following:
y = `root(3)(1 + x^3)`

Exercise 10.3 | Q 5 | Page 163

Differentiate the following:
y = `"e"^sqrt(x)`

Exercise 10.3 | Q 6 | Page 163

Differentiate the following:
y = sin (ex)

Exercise 10.3 | Q 7 | Page 163

Differentiate the following:
F(x) = (x3 + 4x)7

Exercise 10.3 | Q 8 | Page 163

Differentiate the following:

h(t) = `("t" - 1/"t")^(3/2)`

Exercise 10.3 | Q 9 | Page 163

Differentiate the following:

f(t) = `root(3)(1 + tan "t")`

Exercise 10.3 | Q 10 | Page 163

Differentiate the following:
y = cos (a3 + x3)

Exercise 10.3 | Q 11 | Page 163

Differentiate the following:
y = e–mx 

Exercise 10.3 | Q 12 | Page 163

Differentiate the following:
y = 4 sec 5x

Exercise 10.3 | Q 13 | Page 163

Differentiate the following:
y = (2x – 5)4 (8x2 – 5)–3 

Exercise 10.3 | Q 14 | Page 163

Differentiate the following:

y = `(x^2 + 1) root(3)(x^2 + 2)`

Exercise 10.3 | Q 15 | Page 163

Differentiate the following:

y = `x"e"^(-x^2)`

Exercise 10.3 | Q 16 | Page 163

Differentiate the following:

s(t) = `root(4)(("t"^3 + 1)/("t"^3 - 1)`

Exercise 10.3 | Q 17 | Page 163

Differentiate the following:

f(x) = `x/sqrt(7 - 3x)`

Exercise 10.3 | Q 18 | Page 163

Differentiate the following:
y = tan (cos x)

Exercise 10.3 | Q 19 | Page 163

Differentiate the following:

y = `(sin^2x)/cos x`

Exercise 10.3 | Q 20 | Page 163

Differentiate the following:

y = `5^((-1)/x)`

Exercise 10.3 | Q 21 | Page 163

Differentiate the following:
y = `sqrt(1 + 2tanx)`

Exercise 10.3 | Q 22 | Page 164

Differentiate the following:
y = sin3x + cos3x

Exercise 10.3 | Q 23 | Page 164

Differentiate the following:
y = sin2(cos kx)

Exercise 10.3 | Q 24 | Page 164

Differentiate the following:
y = (1 + cos2)6

Exercise 10.3 | Q 25 | Page 164

Differentiate the following:

y = `"e"^(3x)/(1 + "e"^x`

Exercise 10.3 | Q 26 | Page 164

Differentiate the following:
y = `sqrt(x +sqrt(x)`

Exercise 10.3 | Q 27 | Page 164

Differentiate the following:
y = `"e"^(xcosx)`

Exercise 10.3 | Q 28 | Page 164

Differentiate the following:
y = `sqrt(x + sqrt(x + sqrt(x)`

Exercise 10.3 | Q 29 | Page 164

Differentiate the following:
y = `sin(tan(sqrt(sinx)))`

Exercise 10.3 | Q 30 | Page 164

Differentiate the following:

y = `sin^-1 ((1 - x^2)/(1 + x^2))`

Exercise 10.4 [Page 176]

Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 10 Differential Calculus - Differentiability and Methods of DifferentiationExercise 10.4 [Page 176]

(1 - 18) :

Exercise 10.4 | Q 1 | Page 176

Find the derivatives of the following:
y = `x^(cosx)`

Exercise 10.4 | Q 2 | Page 176

Find the derivatives of the following:
y = `x^(logx) + (logx)^x`

Exercise 10.4 | Q 3 | Page 176

Find the derivatives of the following:
`sqrt(x) = "e"^((x - y))`

Exercise 10.4 | Q 4 | Page 176

Find the derivatives of the following:
xy = yx

Exercise 10.4 | Q 5 | Page 176

Find the derivatives of the following:
(cos x)log x

Exercise 10.4 | Q 6 | Page 176

Find the derivatives of the following:

`x^2/"a"^2 + y^2/"b"^2` = 1

Exercise 10.4 | Q 7 | Page 176

Find the derivatives of the following:

`sqrt(x^2 + y^2) = tan^-1 (y/x)`

Exercise 10.4 | Q 8 | Page 176

Find the derivatives of the following:
tan (x + y) + tan (x – y) = x

Exercise 10.4 | Q 9 | Page 176

Find the derivatives of the following:

If cos(xy) = x, show that `(-(1 + ysin(xy)))/(xsiny)`

Exercise 10.4 | Q 10 | Page 176

Find the derivatives of the following:

`tan^-1sqrt((1 - cos x)/(1 + cos x)` 

Exercise 10.4 | Q 11 | Page 176

Find the derivatives of the following:

`tan^-1 = ((6x)/(1 - 9x^2))`

Exercise 10.4 | Q 12 | Page 176

Find the derivatives of the following:

`cos[2tan^-1 sqrt((1 - x)/(1 + x))]`

Exercise 10.4 | Q 13 | Page 176

Find the derivatives of the following:

x = `"a" cos^3"t"` ; y = `"a" sin^3"t"`

Exercise 10.4 | Q 14 | Page 176

Find the derivatives of the following:

x = a (cos t + t sin t); y = a (sin t – t cos t)

Exercise 10.4 | Q 15 | Page 176

Find the derivatives of the following:

x = `(1 - "t"^2)/(1 + "t"^2)`, y = `(2"t")/(1 + "t"^2)`

Exercise 10.4 | Q 16 | Page 176

Find the derivatives of the following:

`cos^-1 ((1 - x^2)/(1 + x^2))`

Exercise 10.4 | Q 17 | Page 176

Find the derivatives of the following:

sin-1 (3x – 4x3)

Exercise 10.4 | Q 18 | Page 176

Find the derivatives of the following:

`tan^-1 ((cos x + sin x)/(cos x - sin x))`

Exercise 10.4 | Q 19 | Page 176

Find the derivatives of the following:

Find the derivative of sin x2 with respect to x2 

Exercise 10.4 | Q 20 | Page 176

Find the derivatives of the following:

Find the derivative of `sin^-1 ((2x)/(1 + x^2))` with respect to `tan^-1 x`

Exercise 10.4 | Q 21 | Page 176

Find the derivatives of the following:

If u = `tan^-1  (sqrt(1 + x^2) - 1)/x` and v = `tan^-1 x`, find `("d"u)/("d"v)`

Exercise 10.4 | Q 22 | Page 176

Find the derivatives of the following:

Find the derivative with `tan^-1 ((sinx)/(1 + cos x))` with respect to `tan^-1 ((cosx)/(1 + sinx))`

Exercise 10.4 | Q 23 | Page 176

Find the derivatives of the following:

If y = sin–1x then find y”

Exercise 10.4 | Q 24 | Page 176

Find the derivatives of the following:

If y = etan–1x, show that (1 + x2)y” + (2x – 1)y’ = 0

Exercise 10.4 | Q 25 | Page 176

Find the derivatives of the following:

If y = `(sin^-1 x)/sqrt(1 - x^2)`, show that (1 – x2)y2 – 3xy1 – y = 0

Exercise 10.4 | Q 26 | Page 176

Find the derivatives of the following:

If x = a(θ + sin θ), y = a(1 – cos θ) then prove that at θ = `pi/2`, yn = `1/"a"`

Exercise 10.4 | Q 27 | Page 176

Find the derivatives of the following:

If sin y = x sin(a + y), the prove that `("d"y)/("d"x) = (sin^2("a" + y))/sin"a"`, a ≠ nπ

Exercise 10.4 | Q 28 | Page 176

Find the derivatives of the following:

If y = `(cos^-1 x)^2`, prove that `(1 - x^2) ("d"^2y)/("d"x)^2 - x ("d"y)/("d"x) - 2` = 0. Hence find y2 when x = 0

Exercise 10.5 [Pages 177 - 179]

Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 10 Differential Calculus - Differentiability and Methods of DifferentiationExercise 10.5 [Pages 177 - 179]

Exercise 10.5 | Q 1 | Page 177

Choose the correct alternative:

`"d"/("d"x) (2/pi sin x^circ)` is

  • `pi/180 cosx^circ`

  • `1/90 cosx^circ`

  • `pi/90 cosx^circ`

  • `2/pi cosx^circ`

Exercise 10.5 | Q 2 | Page 177

Choose the correct alternative:

f y = f(x2 + 2) and f'(3) = 5 , then `("d"y)/("d"x)` at x = 1 is

  • 5

  • 25

  • 15

  • 10

Exercise 10.5 | Q 3 | Page 177

Choose the correct alternative:

If y = `1/4 u^4`, u = `2/3 x^3 + 5`, then `("d"y)/("d"x)` is 

  • `1/27 x^2 (2x^3 + 15)^3`

  • `2/27 x(2x^3 + 5)^3`

  • `2/27 x^2(2x^3 + 15)^3`

  • `- 2/27 x(2x^3 + 5)^3`

Exercise 10.5 | Q 4 | Page 177

Choose the correct alternative:

If f(x) = x2 – 3x, then the points at which f(x) = f’(x) are

  • both positive integers

  • both negative integers

  • both irrational

  • one rational and another irrational

Exercise 10.5 | Q 5 | Page 177

Choose the correct alternative:

If y = `1/("a" - z)`, then `("d"z)/("d"y)` is

  • (a – z)2

  • – (z – a)2

  • (z + a)2

  • – (z + a)2

Exercise 10.5 | Q 6 | Page 177

Choose the correct alternative:

If y = cos (sin x2), then `("d"y)/("d"x)` at x = `sqrt(pi/2)` is

  • – 2

  • 2

  • `- 2 sqrt(pi/2)`

  • 0

Exercise 10.5 | Q 7 | Page 177

Choose the correct alternative:

If y = mx + c and f(0) = f’(0) = 1, then f(2) is

  • 1

  • 2

  • 3

  • – 3

Exercise 10.5 | Q 8 | Page 177

Choose the correct alternative:
If f(x) = x tan-1x then f'(1) is

  • `1 + pi/4`

  • `1/2 + pi/4`

  • `1/2 - pi/4`

  • 2

Exercise 10.5 | Q 9 | Page 177

Choose the correct alternative:

`"d"/("d"x) ("e"^(x + 5log x))` is

  • `"e"^x * x^4 (x + 5)`

  • `"e"^x *x(x + 5)`

  • `"e"^x + 5/x`

  • `"e"^x - 5/x`

Exercise 10.5 | Q 10 | Page 177

Choose the correct alternative:

If the derivative of (ax – 5)e3x at x = 0 is – 13, then the value of a is

  • 8

  • – 2

  • 5

  • 2

Exercise 10.5 | Q 11 | Page 178

Choose the correct alternative:

x = `(1 - "t"^2)/(1 + "t"^2)`, y = `(2"t")/(1 + "t"^2)` then `("d"y)/("d"x)` is

  • `- y/x`

  • `y/x`

  • `- x/y`

  • `x/y`

Exercise 10.5 | Q 12 | Page 178

Choose the correct alternative:

If x = a sin θ and y = b cos θ, then `("d"^2y)/("d"x^2)` is

  • `"a"/"b"^2 sec^2 theta`

  • `- "b"/"a" sec^2 theta`

  • `- "b"/"a"^2 sec^3 theta`

  • `- "b"^2/"a"^2 sec^3 theta`

Exercise 10.5 | Q 13 | Page 178

Choose the correct alternative:

The differential coefficient of `log_10 x` with respect to `log_x 10` is

  • 1

  • `- (log_10 x)^2`

  • `(log_x 10)^2`

  • `x^2/100`

Exercise 10.5 | Q 14 | Page 178

Choose the correct alternative:

If f(x) = x + 2, then f'(f(x)) at x = 4 is

  • 8

  • 1

  • 4

  • 5

Exercise 10.5 | Q 15 | Page 178

Choose the correct alternative:

If y = `(1 - x)^2/x^2`, then `("d"y)/("d"x)` is

  • `2/x^2 + 2/x^3`

  • `- 2/x^2 + 2/x^3`

  • `- 2/x^2 - 2/x^3`

  • `- 2/x^3 + 2/x^2`

Exercise 10.5 | Q 16 | Page 178

Choose the correct alternative:

If pv = 81, then `"dp"/"dv"` at v = 9 is

  • 1

  • – 1

  • 2

  • – 3

Exercise 10.5 | Q 17 | Page 178

Choose the correct alternative:

If f(x) = `{{:(x - 5,  "if"  x ≤ 1),(4x^2 - 9,  "if"  1 < x < 2),(3x + 4,  "if"  x ≥ 2):}` , then the right hand derivative of f(x) at x = 2 is

  • 0

  • 2

  • 3

  • 4

Exercise 10.5 | Q 18 | Page 178

Choose the correct alternative:

It is given that f'(a) exists, then `lim_(x -> "a") (xf("a") - "a"f(x))/(x - "a")` is

  • f(a) – af'(a)

  • f'(a)

  • – f'(a)

  • f(a) + af'(a)

Exercise 10.5 | Q 19 | Page 178

Choose the correct alternative:

If f(x) = `{{:(x + 1,  "when"   x < 2),(2x - 1,  "when"  x ≥ 2):}` , then f'(2) is

  • 0

  • 1

  • 2

  • does not exist

Exercise 10.5 | Q 20 | Page 178

Choose the correct alternative:

If g(x) = (x2 + 2x + 1) f(x) and f(0) = 5 and `lim_(x -> 0) (f(x) - 5)/x` = 4, then g'(0) is

  • 20

  • 14

  • 18

  • 12

Exercise 10.5 | Q 21 | Page 179

Choose the correct alternative:

If f(x) = `{{:(x + 2, - 1 < x < 3),(5, x = 3),(8 - x, x > 3):}` , then at x = 3, f'(x) is

  • 1

  • – 1

  • 0

  • does not exist

Exercise 10.5 | Q 22 | Page 179

Choose the correct alternative:

The derivative of f(x)= x|x| at x = – 3 is

  • 6

  • – 6

  • does not exist

  • 0

Exercise 10.5 | Q 23 | Page 179

Choose the correct alternative:

If f(x) = `{{:(2"a" - x,  "for"  - "a" < x < "a"),(3x - 2"a", "for"  x ≥ "a"):}` , then which one of the following is true?

  • f(x) is not differentiable at x = a

  • f(x) is discontinuous at x = a

  • f(x) is continuous for all x in R

  • f(x) is differentiable for all x ≥ a

Exercise 10.5 | Q 24 | Page 179

Choose the correct alternative:

If f(x) = `{{:("a"x^2 - "b"",", - 1 < x < 1),(1/|x|",",  "elsewhere"):}` is differentiable at x = 1, then

  • a = `1/2`, b = `(-3)/2`

  • a = `(- 1)/2`, b = `3/2`

  • a = `- 1/2`, b = `- 3/2`

  • a = `1/2`, b = `3/2`

Exercise 10.5 | Q 25 | Page 179

Choose the correct alternative:

The number of points in R in which the function f(x) = |x – 1| + |x – 3| + sin x is not differentiable, is

  • 3

  • 2

  • 1

  • 4

Chapter 10: Differential Calculus - Differentiability and Methods of Differentiation

Exercise 10.1Exercise 10.2Exercise 10.3Exercise 10.4Exercise 10.5
Class 11th Mathematics Volume 1 and 2 Answers Guide - Shaalaa.com

Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide chapter 10 - Differential Calculus - Differentiability and Methods of Differentiation

Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide chapter 10 (Differential Calculus - Differentiability and Methods of Differentiation) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the Tamil Nadu Board of Secondary Education Class 11th Mathematics Volume 1 and 2 Answers Guide solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 11th Mathematics Volume 1 and 2 Answers Guide chapter 10 Differential Calculus - Differentiability and Methods of Differentiation are Introduction of Differential Calculus-differentiability and Methods of Differentiation, The Concept of Derivative, Differentiability and Continuity, Differentiation Rules.

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