# Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board chapter 3 - Differentiation [Latest edition]

## Chapter 3: Differentiation

Exercise 3.1Exercise 3.2Exercise 3.3Exercise 3.4Exercise 3.5Exercise 3.6Miscellaneous Exercise 3
Exercise 3.1 [Pages 90 - 91]

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 3 Differentiation Exercise 3.1 [Pages 90 - 91]

Exercise 3.1 | Q 1.1 | Page 91

Find "dy"/"dx" if, y = sqrt("x" + 1/"x")

Exercise 3.1 | Q 1.2 | Page 90

Find "dy"/"dx" if, y = root(3)("a"^2 + "x"^2)

Exercise 3.1 | Q 1.3 | Page 91

Find "dy"/"dx" if, y = (5x3 - 4x2 - 8x)9

Exercise 3.1 | Q 2.1 | Page 91

Find "dy"/"dx" if, y = log(log x)

Exercise 3.1 | Q 2.2 | Page 91

Find "dy"/"dx" if, y = log(10x4 + 5x3 - 3x2 + 2)

Exercise 3.1 | Q 2.3 | Page 91

Find "dy"/"dx" if, y = log(ax2 + bx + c)

Exercise 3.1 | Q 3.1 | Page 91

Find "dy"/"dx" if, y = "e"^(5"x"^2 - 2"x" + 4)

Exercise 3.1 | Q 3.2 | Page 91

Find "dy"/"dx" if, y = "a"^((1 + log "x"))

Exercise 3.1 | Q 3.3 | Page 91

Find "dy"/"dx" if, y = 5^(("x" + log"x"))

Exercise 3.2 [Page 92]

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 3 Differentiation Exercise 3.2 [Page 92]

Exercise 3.2 | Q 1.1 | Page 92

Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = 12 + 10x + 25x2

Exercise 3.2 | Q 1.2 | Page 92

Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = 18x + log(x - 4).

Exercise 3.2 | Q 1.3 | Page 92

Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = 25x + log(1 + x2)

Exercise 3.2 | Q 2.1 | Page 92

Find the marginal demand of a commodity where demand is x and price is y.

y = "x"*"e"^-"x" + 7

Exercise 3.2 | Q 2.2 | Page 92

Find the marginal demand of a commodity where demand is x and price is y.

y = ("x + 2")/("x"^2 + 1)

Exercise 3.2 | Q 2.3 | Page 92

Find the marginal demand of a commodity where demand is x and price is y.

y = (5"x" + 9)/(2"x" - 10)

Exercise 3.3 [Page 94]

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 3 Differentiation Exercise 3.3 [Page 94]

Exercise 3.3 | Q 1.1 | Page 94

Find "dy"/"dx"if, y = "x"^("x"^"2x")

Exercise 3.3 | Q 1.2 | Page 94

Find "dy"/"dx"if, y = "x"^("e"^"x")

Exercise 3.3 | Q 1.3 | Page 94

Find "dy"/"dx"if, y = "e"^("x"^"x")

Exercise 3.3 | Q 2.1 | Page 94

Find "dy"/"dx"if, y = (1 + 1/"x")^"x"

Exercise 3.3 | Q 2.2 | Page 94

Find "dy"/"dx"if, y = (2x + 5)x

Exercise 3.3 | Q 2.3 | Page 94

Find "dy"/"dx"if, y = root(3)(("3x" - 1)/(("2x + 3")(5 - "x")^2))

Exercise 3.3 | Q 3.1 | Page 94

Find "dy"/"dx"if, y = (log "x"^"x") + "x"^(log "x")

Exercise 3.3 | Q 3.2 | Page 94

Find "dy"/"dx"if, y = ("x")^"x" + ("a"^"x")

Exercise 3.3 | Q 3.3 | Page 94

Find "dy"/"dx"if, y = 10^("x"^"x") + 10^("x"^10) + 10^(10^"x")

Exercise 3.4 [Page 95]

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 3 Differentiation Exercise 3.4 [Page 95]

Exercise 3.4 | Q 1.1 | Page 95

Find "dy"/"dx" if, sqrt"x" + sqrt"y" = sqrt"a"

Exercise 3.4 | Q 1.2 | Page 95

Find "dy"/"dx" if, x3 + y3 + 4x3y = 0

Exercise 3.4 | Q 1.3 | Page 95

Find "dy"/"dx" if, x3 + x2y + xy2 + y3 = 81

Exercise 3.4 | Q 2.1 | Page 95

Find "dy"/"dx" if, yex + xey = 1

Exercise 3.4 | Q 2.2 | Page 95

Find "dy"/"dx" if, "x"^"y" = "e"^("x - y")

Exercise 3.4 | Q 2.3 | Page 95

Find "dy"/"dx" if, xy = log (xy)

Exercise 3.4 | Q 3.1 | Page 95

Solve the following:

If "x"^5 * "y"^7 = ("x + y")^12 then show that, "dy"/"dx" = "y"/"x"

Exercise 3.4 | Q 3.2 | Page 95

Solve the following:

If log (x + y) = log (xy) + a then show that, "dy"/"dx" = (- "y"^2)/"x"^2.

Exercise 3.4 | Q 3.3 | Page 95

Solve the following:

If "e"^"x" + "e"^"y" = "e"^("x + y") then show that, "dy"/"dx" = - "e"^"y - x".

Exercise 3.5 [Page 97]

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 3 Differentiation Exercise 3.5 [Page 97]

Exercise 3.5 | Q 1.1 | Page 97

Find "dy"/"dx", if x = at2, y = 2at

Exercise 3.5 | Q 1.2 | Page 97

Find "dy"/"dx", if x = 2at2 , y = at4

Exercise 3.5 | Q 1.3 | Page 97

Find "dy"/"dx", if x = e3t , y = "e"^(4"t" + 5)

Exercise 3.5 | Q 2.1 | Page 97

Find "dy"/"dx", if x = ("u" + 1/"u")^2, "y" = (2)^(("u" + 1/"u"))

Exercise 3.5 | Q 2.2 | Page 97

Find "dy"/"dx", if x = sqrt(1 + "u"^2), "y" = log (1 + "u"^2)

Exercise 3.5 | Q 2.3 | Page 97

Find "dy"/"dx", if Differentiate 5x with respect to log x

Exercise 3.5 | Q 3.1 | Page 97

Solve the following.

If x = "a"(1 - 1/"t"), "y" = "a"(1 + 1/"t"), then show that "dy"/"dx" = - 1

Exercise 3.5 | Q 3.2 | Page 97

Solve the following.

If x = "4t"/(1 + "t"^2), "y" = 3((1 - "t"^2)/(1 + "t"^2)) then show that "dy"/"dx" = (-9"x")/"4y".

Exercise 3.5 | Q 3.3 | Page 97

Solve the following.

If x = t . log t, y = tt, then show that "dy"/"dx" - "y" = 0

Exercise 3.6 [Page 98]

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 3 Differentiation Exercise 3.6 [Page 98]

Exercise 3.6 | Q 1.1 | Page 98

Find ("d"^2"y")/"dx"^2, if y = sqrt"x"

Exercise 3.6 | Q 1.2 | Page 98

Find ("d"^2"y")/"dx"^2, if y = "x"^5

Exercise 3.6 | Q 1.3 | Page 98

Find ("d"^2"y")/"dx"^2, if y = "x"^-7

Exercise 3.6 | Q 2.1 | Page 98

Find ("d"^2"y")/"dx"^2, if y = "e"^"x"

Exercise 3.6 | Q 2.2 | Page 98

Find ("d"^2"y")/"dx"^2, if y = "e"^((2"x" + 1))

Exercise 3.6 | Q 2.3 | Page 98

Find ("d"^2"y")/"dx"^2, if y = "e"^"log x"

Miscellaneous Exercise 3 [Pages 99 - 101]

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 3 Differentiation Miscellaneous Exercise 3 [Pages 99 - 101]

Miscellaneous Exercise 3 | Q 1.01 | Page 99

Choose the correct alternative.

If y = (5x3 - 4x2 - 8x)9, then "dy"/"dx" =

• 9(5x3 - 4x2 - 8x)8 (15x2 - 8x - 8)

• 9(5x3 - 4x2 - 8x)9 (15x2 - 8x - 8)

• 9(5x3 - 4x2 - 8x)8 (5x2 - 8x - 8)

• 9(5x3 - 4x2 - 8x)9 (15x2 - 8x - 8)

Miscellaneous Exercise 3 | Q 1.02 | Page 99

Choose the correct alternative.

If y = sqrt("x" + 1/"x"), then "dy"/"dx" = ?

• ("x"^2 - 1)/(2"x"^2sqrt("x"^2 + 1))

• (1 - "x"^2)/(2"x"^2("x"^2 + 1))

• ("x"^2 - 1)/("2x"sqrt"x"sqrt("x"^2 + 1))

• (1 - "x"^2)/("2x"sqrt"x"sqrt("x"^2 + 1))

Miscellaneous Exercise 3 | Q 1.03 | Page 99

Choose the correct alternative.

If y = "e"^(log "x"), then "dy"/"dx" = ?

• ("e"^(log "x"))/"x"

• 1/"x"

• 0

• 1/2

Miscellaneous Exercise 3 | Q 1.04 | Page 99

Choose the correct alternative.

If y = 2x2 + 22 + a2, then "dy"/"dx" = ?

• x

• 4x

• 2x

• -2x

Miscellaneous Exercise 3 | Q 1.05 | Page 99

Choose the correct alternative.

If y = 5x . x5, then "dy"/"dx" = ?

• 5x. x(5 + log 5)

• 5x. x(5 + log 5)

• 5x . x(5 + x log 5)

• 5x. x(5 + x log 5)

Miscellaneous Exercise 3 | Q 1.06 | Page 99

Choose the correct alternative.

If y = log ("e"^"x"/"x"^2), then "dy"/"dx" = ?

• (2 - "x")/"x"

• ("x" - 2)/"x"

• ("e - x")/"ex"

• ("x - e")/"ex"

Miscellaneous Exercise 3 | Q 1.07 | Page 99

Choose the correct alternative.

If ax2 + 2hxy + by2 = 0 then "dy"/"dx" = ?

• (("ax" + "hx"))/(("hx" + "by"))

• (-("ax" + "hx"))/(("hx" + "by"))

• (("ax" - "hx"))/(("hx" + "by"))

• (("2ax" + "hy"))/(("hx" + "3by"))

Miscellaneous Exercise 3 | Q 1.08 | Page 99

Choose the correct alternative.

If "x"^4."y"^5 = ("x + y")^("m + 1") then "dy"/"dx" = "y"/"x" then m = ?

• 8

• 4

• 5

• 20

Miscellaneous Exercise 3 | Q 1.09 | Page 99

Choose the correct alternative.

If x = ("e"^"t" + "e"^-"t")/2, "y" = ("e"^"t" - "e"^-"t")/2  then "dy"/"dx" = ?

• "-y"/"x"

• "y"/"x"

• "-x"/"y"

• "x"/"y"

Miscellaneous Exercise 3 | Q 1.1 | Page 99

Choose the correct alternative.

If x = 2at2 , y = 4at, then "dy"/"dx" = ?

• - 1/(2"at"^2)

• 1/(2"at"^3)

• 1/"t"

• 1/"4at"^3

Miscellaneous Exercise 3 | Q 2.01 | Page 99

Fill in the Blank

If 3x2y + 3xy2 = 0, then "dy"/"dx" = ________

Miscellaneous Exercise 3 | Q 2.02 | Page 99

Fill in the Blank

If "x"^"m"*"y"^"n" = ("x + y")^("m + n"), then "dy"/"dx" = square/"x"

Miscellaneous Exercise 3 | Q 2.03 | Page 99

Fill in the Blank

If 0 = log(xy) + a, then "dy"/"dx" =  (-"y")/square

Miscellaneous Exercise 3 | Q 2.04 | Page 99

Fill in the blank.

If x = t log t and y = tt, then "dy"/"dx" = ____

Miscellaneous Exercise 3 | Q 2.05 | Page 99

Fill in the blank.

If y = x . log x, then ("d"^2"y")/"dx"^2= _____

Miscellaneous Exercise 3 | Q 2.06 | Page 100

Fill in the blank.

If y = [log ("x")]^2  "then" ("d"^2"y")/"dx"^2 = _____

Miscellaneous Exercise 3 | Q 2.07 | Page 100

Fill in the blank.

If x = "y" + 1/"y", then "dy"/"dx" =____

Miscellaneous Exercise 3 | Q 2.08 | Page 100

Fill in the blank.

If y = "e"^"ax", then "x" * "dy"/"dx" =____

Miscellaneous Exercise 3 | Q 2.09 | Page 100

Fill in the blank.

If x = t log t and y = tt, then "dy"/"dx" = ____

Miscellaneous Exercise 3 | Q 2.1 | Page 100

Fill in the blank.

If y = ("x" + sqrt("x"^2 - 1))^"m", then ("x"^2 - 1) "dy"/"dx" = ______

Miscellaneous Exercise 3 | Q 3.1 | Page 100

State whether the following is True or False:

If f′ is the derivative of f, then the derivative of the inverse of f is the inverse of f′.

• True

• False

Miscellaneous Exercise 3 | Q 3.2 | Page 100

State whether the following is True or False:

The derivative of log_"a""x", where a is constant is 1/("x"*log"a").

• True

• False

Miscellaneous Exercise 3 | Q 3.3 | Page 100

State whether the following is True or False:

The derivative of f(x) = ax, where a is constant is x.ax-1.

• True

• False

Miscellaneous Exercise 3 | Q 3.4 | Page 100

State whether the following is True or False:

The derivative of polynomial is polynomial.

• True

• False

Miscellaneous Exercise 3 | Q 3.5 | Page 100

State whether the following is True or False:

"d"/"dx"(10^"x") = "x"*10^("x" - 1)

• True

• False

Miscellaneous Exercise 3 | Q 3.6 | Page 100

State whether the following is True or False:

If y = log x, then "dy"/"dx" = 1/"x"

• True

• False

Miscellaneous Exercise 3 | Q 3.7 | Page 100

State whether the following is True or False:

If y = e2, then "dy"/"dx" = 2"e"

• True

• False

Miscellaneous Exercise 3 | Q 3.8 | Page 100

State whether the following is True or False:

The derivative of ax is ax . loga.

• True

• False

Miscellaneous Exercise 3 | Q 3.9 | Page 100

State whether the following is True or False:

The derivative of "x"^"m"*"y"^"n" = ("x + y")^("m + n") is "x"/"y"

• True

• False

Miscellaneous Exercise 3 | Q 4.01 | Page 100

Solve the following:

If y = (6x3 - 3x2 - 9x)10, find "dy"/"dx"

Miscellaneous Exercise 3 | Q 4.02 | Page 100

Solve the following:

If y = root(5)((3"x"^2 + 8"x" + 5)^4), find "dy"/"dx"

Miscellaneous Exercise 3 | Q 4.03 | Page 100

Solve the following:

If y = [log(log(logx))]2, find "dy"/"dx"

Miscellaneous Exercise 3 | Q 4.04 | Page 100

Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = 25 + 30x  – x2.

Miscellaneous Exercise 3 | Q 4.05 | Page 100

Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = (5"x" + 7)/("2x" - 13).

Miscellaneous Exercise 3 | Q 4.06 | Page 100

Find "dy"/"dx", if y = xx.

Miscellaneous Exercise 3 | Q 4.07 | Page 100

Find "dy"/"dx", if y = 2^("x"^"x").

Miscellaneous Exercise 3 | Q 4.08 | Page 100

Find "dy"/"dx" if y = sqrt(((3"x" - 4)^3)/(("x + 1")^4("x + 2")))

Miscellaneous Exercise 3 | Q 4.09 | Page 100

Find "dy"/"dx" if y = "x"^"x" + ("7x" - 1)^"x"

Miscellaneous Exercise 3 | Q 4.1 | Page 100

If y = "x"^3 + 3"xy"^2 + 3"x"^2"y" Find "dy"/"dx"

Miscellaneous Exercise 3 | Q 4.11 | Page 100

If "x"^3 + "y"^2 + "xy" = 7 Find "dy"/"dx"

Miscellaneous Exercise 3 | Q 4.12 | Page 100

If "x"^3"y"^3 = "x"^2 - "y"^2, Find "dy"/"dx"

Miscellaneous Exercise 3 | Q 4.13 | Page 100

If "x"^7*"y"^9 = ("x + y")^16, then show that "dy"/"dx" = "y"/"x"

Miscellaneous Exercise 3 | Q 4.14 | Page 100

If "x"^"a"*"y"^"b" = ("x + y")^("a + b"), then show that "dy"/"dx" = "y"/"x"

Miscellaneous Exercise 3 | Q 4.15 | Page 100

Find "dy"/"dx" if x = 5t2, y = 10t.

Miscellaneous Exercise 3 | Q 4.16 | Page 100

Find "dy"/"dx" if x = "e"^"3t",  "y" = "e"^(sqrt"t").

Miscellaneous Exercise 3 | Q 4.17 | Page 100

Differentiate log (1 + x2) with respect to ax.

Miscellaneous Exercise 3 | Q 4.18 | Page 101

Differentiate "e"^("4x" + 5) with respect to 104x.

Miscellaneous Exercise 3 | Q 4.19 | Page 101

Find ("d"^2"y")/"dx"^2, if y = log (x).

Miscellaneous Exercise 3 | Q 4.2 | Page 101

Find ("d"^2"y")/"dx"^2, if y = 2at, x = at2

Miscellaneous Exercise 3 | Q 4.21 | Page 101

Find ("d"^2"y")/"dx"^2, if y = "x"^2 * "e"^"x"

Miscellaneous Exercise 3 | Q 4.22 | Page 101

If x2 + 6xy + y2 = 10, then show that ("d"^2"y")/"dx"^2 = 80/("3x" + "y")^3.

Miscellaneous Exercise 3 | Q 4.23 | Page 101

If ax2 + 2hxy + by2 = 0, then show that ("d"^2"y")/"dx"^2 = 0

## Chapter 3: Differentiation

Exercise 3.1Exercise 3.2Exercise 3.3Exercise 3.4Exercise 3.5Exercise 3.6Miscellaneous Exercise 3

## Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board chapter 3 - Differentiation

Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board chapter 3 (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 Maharashtra State Board Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. Balbharati textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board chapter 3 Differentiation are Derivatives of Composite Functions - Chain Rule, Derivatives of Inverse Functions, Derivatives of Logarithmic Functions, Derivatives of Implicit Functions, Derivatives of Parametric Functions, Second Order Derivative.

Using Balbharati 12th Board Exam solutions Differentiation exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in Balbharati Solutions are important questions that can be asked in the final exam. Maximum students of Maharashtra State Board 12th Board Exam prefer Balbharati Textbook Solutions to score more in exam.

Get the free view of chapter 3 Differentiation 12th Board Exam extra questions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board and can use Shaalaa.com to keep it handy for your exam preparation