Tamil Nadu Board of Secondary EducationHSC Science Class 11th

# 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]

#### Chapters ## Chapter 10: Differential Calculus - Differentiability and Methods of Differentiation

Exercise 10.1Exercise 10.2Exercise 10.3Exercise 10.4Exercise 10.5
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 Differentiation Exercise 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 Differentiation Exercise 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 Differentiation Exercise 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 Differentiation Exercise 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 Differentiation Exercise 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 ## 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

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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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