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## Chapter 3: Differentiation

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

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

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

Find `"dy"/"dx"` if, y = (5x^{3} - 4x^{2} - 8x)^{9}

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

Find `"dy"/"dx"` if, y = log(10x^{4} + 5x^{3} - 3x^{2} + 2)

Find `"dy"/"dx"` if, y = log(ax^{2} + bx + c)

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

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

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

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

Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = 12 + 10x + 25x^{2}

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

Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = 25x + log(1 + x^{2})

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

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

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

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

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

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

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

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

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

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

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

Find `"dy"/"dx"`if, y = (2x + 5)^{x}

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

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

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

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

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

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

Find `"dy"/"dx"` if, x^{3} + y^{3} + 4x^{3}y = 0

Find `"dy"/"dx"` if, x^{3} + x^{2}y + xy^{2} + y^{3} = 81

Find `"dy"/"dx"` if, ye^{x} + xe^{y} = 1

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

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

**Solve the following:**

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

**Solve the following:**

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

**Solve the following:**

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

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

Find `"dy"/"dx"`, if x = at^{2}, y = 2at

Find `"dy"/"dx"`, if x = 2at^{2} , y = at^{4}

Find `"dy"/"dx"`, if x = e^{3t} , y = `"e"^(4"t" + 5)`

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

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

Find `"dy"/"dx"`, if Differentiate 5^{x}^{ }with respect to log x

**Solve the following.**

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

**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"`.

**Solve the following.**

If x = t . log t, y = t^{t}, then show that `"dy"/"dx" - "y" = 0`

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

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

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

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

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

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

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

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

**Choose the correct alternative.**

If y = (5x^{3} - 4x^{2} - 8x)^{9}, then `"dy"/"dx"` =

9(5x

^{3}- 4x^{2}- 8x)^{8}(15x^{2}- 8x - 8)9(5x

^{3}- 4x^{2}- 8x)^{9}(15x^{2}- 8x - 8)9(5x

^{3}- 4x^{2}- 8x)^{8}(5x^{2}- 8x - 8)9(5x

^{3}- 4x^{2}- 8x)^{9}(15x^{2}- 8x - 8)

**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))`

**Choose the correct alternative.**

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

`("e"^(log "x"))/"x"`

`1/"x"`

0

`1/2`

**Choose the correct alternative.**

If y = 2x^{2} + 2^{2} + a^{2}, then `"dy"/"dx" = ?`

x

4x

2x

-2x

**Choose the correct alternative.**

If y = 5^{x} . x^{5}, then `"dy"/"dx" = ?`

5

^{x}. x^{4 }(5 + log 5)5

^{x}. x^{5 }(5 + log 5)5

^{x}. x^{4 }(5 + x log 5)5

^{x}. x^{5 }(5 + x log 5)

**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"`

**Choose the correct alternative.**

If ax^{2} + 2hxy + by^{2} = 0 then `"dy"/"dx" = ?`

`(("ax" + "hx"))/(("hx" + "by"))`

`(-("ax" + "hx"))/(("hx" + "by"))`

`(("ax" - "hx"))/(("hx" + "by"))`

`(("2ax" + "hy"))/(("hx" + "3by"))`

**Choose the correct alternative.**

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

8

4

5

20

**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"`

**Choose the correct alternative.**

If x = 2at^{2} , y = 4at, then `"dy"/"dx" = ?`

`- 1/(2"at"^2)`

`1/(2"at"^3)`

`1/"t"`

`1/"4at"^3`

**Fill in the Blank**

If 3x^{2}y + 3xy^{2} = 0, then `"dy"/"dx"` = ________

**Fill in the Blank**

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

**Fill in the Blank**

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

**Fill in the blank.**

If x = t log t and y = t^{t}, then `"dy"/"dx"` = ____

**Fill in the blank.**

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

**Fill in the blank.**

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

**Fill in the blank.**

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

**Fill in the blank.**

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

**Fill in the blank.**

If x = t log t and y = t^{t}, then `"dy"/"dx"` = ____

**Fill in the blank.**

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

**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

**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

**State whether the following is True or False:**

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

True

False

**State whether the following is True or False:**

The derivative of polynomial is polynomial.

True

False

**State whether the following is True or False:**

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

True

False

**State whether the following is True or False:**

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

True

False

**State whether the following is True or False:**

If y = e^{2}, then `"dy"/"dx" = 2"e"`

True

False

**State whether the following is True or False:**

The derivative of a^{x} is a^{x} . loga.

True

False

**State whether the following is True or False:**

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

True

False

**Solve the following:**

If y = (6x^{3} - 3x^{2} - 9x)^{10}, find `"dy"/"dx"`

**Solve the following:**

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

**Solve the following:**

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

Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = 25 + 30x – x^{2}.

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

Find `"dy"/"dx"`, if y = x^{x}.

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

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

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

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

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

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

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

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

Find `"dy"/"dx"` if x = 5t^{2}, y = 10t.

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

Differentiate log (1 + x^{2}) with respect to a^{x}.

Differentiate `"e"^("4x" + 5)` with respect to 10^{4x}.

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

Find `("d"^2"y")/"dx"^2`, if y = 2at, x = at^{2}

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

If x^{2} + 6xy + y^{2} = 10, then show that `("d"^2y)/("d"x^2) = 80/(3x + y)^3`

If ax^{2} + 2hxy + by^{2} = 0, then show that `("d"^2"y")/"dx"^2` = 0

## Chapter 3: Differentiation

## 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.

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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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