Chapters
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 = (5x3 - 4x2 - 8x)9
Find `"dy"/"dx"` if, y = log(log x)
Find `"dy"/"dx"` if, y = log(10x4 + 5x3 - 3x2 + 2)
Find `"dy"/"dx"` if, y = log(ax2 + 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 + 25x2
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 + x2)
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, x3 + y3 + 4x3y = 0
Find `"dy"/"dx"` if, x3 + x2y + xy2 + y3 = 81
Find `"dy"/"dx"` if, yex + xey = 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 = at2, y = 2at
Find `"dy"/"dx"`, if x = 2at2 , y = at4
Find `"dy"/"dx"`, if x = e3t , 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 5x 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 = tt, 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 = (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)
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 = 2x2 + 22 + a2, then `"dy"/"dx" = ?`
x
4x
2x
-2x
Choose the correct alternative.
If y = 5x . x5, then `"dy"/"dx" = ?`
5x. x4 (5 + log 5)
5x. x5 (5 + log 5)
5x . x4 (5 + x log 5)
5x. x5 (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 ax2 + 2hxy + by2 = 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 = 2at2 , y = 4at, then `"dy"/"dx" = ?`
`- 1/(2"at"^2)`
`1/(2"at"^3)`
`1/"t"`
`1/"4at"^3`
Fill in the Blank
If 3x2y + 3xy2 = 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 = tt, 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 = tt, 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) = ax, where a is constant is x.ax-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 = e2, then `"dy"/"dx" = 2"e"`
True
False
State whether the following is True or False:
The derivative of ax is ax . 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 = (6x3 - 3x2 - 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 – x2.
Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = `(5"x" + 7)/("2x" - 13)`.
Find `"dy"/"dx"`, if y = xx.
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 = 5t2, y = 10t.
Find `"dy"/"dx"` if x = `"e"^"3t", "y" = "e"^(sqrt"t")`.
Differentiate log (1 + x2) with respect to ax.
Differentiate `"e"^("4x" + 5)` with respect to 104x.
Find `("d"^2"y")/"dx"^2`, if y = log (x).
Find `("d"^2"y")/"dx"^2`, if y = 2at, x = at2
Find `("d"^2"y")/"dx"^2`, if y = `"x"^2 * "e"^"x"`
If x2 + 6xy + y2 = 10, then show that `("d"^2"y")/"dx"^2 = 80/("3x" + "y")^3`.
If ax2 + 2hxy + by2 = 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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