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

#### Chapters ## 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 = (5x + 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"^2y)/("d"x^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

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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.

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