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Chapters
1.2: Matrices
▶ 1.3: Differentiation
1.4: Applications of Derivatives
1.5: Integration
1.6: Definite Integration
1.7: Application of Definite Integration
1.8: Differential Equation and Applications
2.1: Commission, Brokerage and Discount
2.2: Insurance and Annuity
2.3: Linear Regression
2.4: Time Series
2.5: Index Numbers
2.6: Linear Programming
2.7: Assignment Problem and Sequencing
2.8: Probability Distributions
![SCERT Maharashtra solutions for Mathematics and Statistics (Commerce) [English] 12 Standard HSC chapter 1.3 - Differentiation - Shaalaa.com](/images/mathematics-and-statistics-commerce-english-12-standard-hsc_6:5f2b1b2038084cf381bfa42c826a928c.jpg "SCERT Maharashtra solutions for Mathematics and Statistics (Commerce) [English] 12 Standard HSC chapter 1.3 - Differentiation")
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Solutions for Chapter 1.3: Differentiation
Below listed, you can find solutions for Chapter 1.3 of Maharashtra State Board SCERT Maharashtra for Mathematics and Statistics (Commerce) [English] 12 Standard HSC.
SCERT Maharashtra solutions for Mathematics and Statistics (Commerce) [English] 12 Standard HSC 1.3 Differentiation Q.1
MCQ [1 Mark]
If y = `1/sqrt(3x^2 - 2x - 1)`, then `("d"y)/("d"x)` = ?
`(-2)/3 (3x - 2) (3x^2 - 2x - 1)^((-3)/2)`
`(-3)/2 (3x - 2) (3x^2 - 2x - 1)^((-3)/2)`
`(3x - 1) (3x^2 - 2x - 1)^((-3)/2)`
`-(3x - 1) (3x^2 - 2x - 1)^((-3)/2)`
Choose the correct alternative:
If y = `root(3)((3x^2 + 8x - 6)^5`, then `("d"y)/("d"x)` = ?
`5/3 (6x + 8) (3x^2 + 8x - 6)^(2/3)`
`(-5)/3 (6x + 8) (3x^2 + 8x - 6)^(2/3)`
`3/5 (3x + 4) (3x^2 + 8x - 6)^(2/3)`
`(-3)/5 (3x + 4) (3x^2 + 8x - 6)^(2/3)`
Choose the correct alternative:
What is the rate of change of demand (x) of a commodity with respect to its price (y) if y = 10 + x + 25x3.
`10/(1 + 75x^2)`
`1/(1 + 75x^2)`
1 + 75x2
`(-1)/(1 + 75x^2)`
Choose the correct alternative:
What is the rate of change of demand (x) of a commodity with respect to its price (y) if y = `(3x + 7)/(2x^2 + 5)`
`(2x^2 + 5)^2/((-6x^2 - 38x + 15))`
`(2x^2 + 5)^2/((-6x^2 - 28x + 15))`
`(2x^2 + 5)^2/((6x^2 - 28x + 15))`
`(2x^2 + 5)^2/((6x^2 - 38x + 15))`
Choose the correct alternative:
If y = `x^(sqrt(x))`, then `("d"y)/("d"x)` = ?
`y/(2sqrt(x))(log x + 2)`
`y/sqrt(x)(log x + 2)`
`y/(2sqrt(x))(log x - 2)`
`y/sqrt(x)(log x - 2)`
Choose the correct alternative:
If y = (x )x + (10)x, then `("d"y)/("d"x)` = ?
xx(1 – log x) + 10xlog10
xx(1 + log x) – 10xlog10
x(1 + log x) + 10xlog10
xx(1 + log x) + 10xlog10
Choose the correct alternative:
If xm. yn = `("x" + "y")^(("m" + "n"))`, then `("dy")/("dx")` = ?
`"y"/"x"`
`(-"y")/"x"`
`"x"/"y"`
`(-"x")/"y"`
If xy = 2x – y, then `("d"y)/("d"x)` = ______
`(xlog2 - y)/(xlog2x)`
`(xlog2 + y)/(xlog2x)`
`(xlog2 + x)/(ylog2x)`
`(ylog2 - x)/(xlog2x)`
Choose the correct alternative:
If x = 2am, y = 2am2, where m be the parameter, then `("d"y)/("d"x)` = ?
2m
– 2m
– am
am
If x = `"a"("t" - 1/"t")`, y = `"a"("t" + 1/"t")`, where t be the parameter, then `("d"y)/("d"x)` = ?
`x/y`
`(-x)/y`
`y/x`
`(-y)/x`
Choose the correct alternative:
If x = at2, y = 2at, then `("d"^2y)/("d"x^2)` = ?
`1/("at"^3)`
`(-1)/(2"at"^3)`
`(-1)/("at"^2)`
`1/("at"^2)`
SCERT Maharashtra solutions for Mathematics and Statistics (Commerce) [English] 12 Standard HSC 1.3 Differentiation Q.2
Fill in the blanks: [1 Mark]
If y = (5x3 – 4x2 – 8x)9, then `("d"y)/("d"x)` is ______
If y = `"a"^((1 + log"x"))`, then `("d"y)/("d"x)` is ______
The rate of change of demand (x) of a commodity with respect to its price (y) is ______ if y = 5 + x2e–x + 2x
The rate of change of demand (x) of a commodity with respect to its price (y) is ______ if y = xe–x + 7
If y = x10, then `("d"y)/("d"x)` is ______
If y = `("e")^((2x + 5))`, then `("d"y)/("d"x)` is ______
If `sqrt(x) + sqrt(y) = sqrt("a")`, then `("d"y)/("d"x)` is ______
If u = 5x and v = log x, then `("du")/("dv")` is ______
If u = ex and v = loge x, then `("du")/("dv")` is ______
If y = x2, then `("d"^2y)/("d"x^2)` is ______
SCERT Maharashtra solutions for Mathematics and Statistics (Commerce) [English] 12 Standard HSC 1.3 Differentiation Q.3
[1 Mark]
State whether the following statement is True or False:
If y = log(log x), then `("d"y)/("d"x)` = logx
True
False
State whether the following statement is True or False:
If y = 10x + 1, then `("d"y)/("d"x)` = 10x.log10
True
False
State whether the following statement is True or False:
If y = x2, then the rate of change of demand (x) of a commodity with respect to its price (y) is `1/(2x)`
True
False
State whether the following statement is True or False:
If y = 7x + 1, then the rate of change of demand (x) of a commodity with respect to its price (y) is 7
True
False
State whether the following statement is True or False:
If y = ex, then `("d"y)/("d"x)` = ex
True
False
State whether the following statement is True or False:
If y = 4x, then `("d"y)/("d"x)` = 4x
True
False
State whether the following statement is True or False:
If `sqrt(x) + sqrt(y) = sqrt("a")`, then `("d"y)/("d"x) = 1/(2sqrt(x)) + 1/(2sqrt(y)) = 1/(2sqrt("a"))`
True
False
State whether the following statement is True or False:
If x2 + y2 = a2, then `("d"y)/("d"x)` = = 2x + 2y = 2a
True
False
State whether the following statement is True or False:
If x = 2at, y = 2a, where t is parameter, then `("d"y)/("d"x) = 1/"t"`
True
False
State whether the following statement is True or False:
If x = 5m, y = m, where m is parameter, then `("d"y)/("d"x) = 1/5`
True
False
State whether the following statement is True or False:
If y = ex, then `("d"^2y)/("d"x^2)` = ex
True
False
SCERT Maharashtra solutions for Mathematics and Statistics (Commerce) [English] 12 Standard HSC 1.3 Differentiation Q.4
Solve the following: [3 Marks]
Find `("d"y)/("d"x)`, if y = [log(log(logx))]2
Find `("d"y)/("d"x)`, if y = (6x3 – 3x2 – 9x)10
Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = `(5x + 7)/(2x - 13)`
Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = 5 + x2e–x + 2x
Find `(dy)/(dx)`, if xy = yx
Find `("d"y)/("d"x)`, if xy = log(xy)
Find `("d"y)/("d"x)`, if x = `sqrt(1 + "u"^2)`, y = log(1 +u2)
If x = t.logt, y = tt, then show that `("d"y)/("d"x)` = tt
Find `("d"^2y)/("d"x^2)`, if y = `"e"^((2x + 1))`
SCERT Maharashtra solutions for Mathematics and Statistics (Commerce) [English] 12 Standard HSC 1.3 Differentiation Q.5
Solve the following : [4 Marks]
Find `("d"y)/("d"x)`, if y = (log x)x + (x)logx
Find `("d"y)/("d"x)`, if y = `root(5)((3x^2 + 8x + 5)^4`
Find `("d"y)/("d"x)`, if y = xx + (7x – 1)x
Find rate of change of demand (x) of a commodity with respect to its price (y) if y = `(3x + 7)/(2x^2 + 5)`
Find `("d"y)/("d"x)`, if y = `x^(x^x)`
Find `("d"y)/("d"x)`, if y = `root(3)(((3x - 1))/((2x + 3)(5 - x)^2)`
Solve the following:
If `"x"^5 * "y"^7 = ("x + y")^12` then show that, `"dy"/"dx" = "y"/"x"`
If xa .yb = `(x + y)^((a + b))`, then show that `("d"y)/("d"x) = y/x`
If x = `(4"t")/(1 + "t"^2)`, y = `3((1 - "t"^2)/(1 + "t"^2))`, then show that `("d"y)/("d"x) = (-9x)/(4y)`
If x2 + 6xy + y2 = 10, then show that `("d"^2y)/("d"x^2) = 80/(3x + y)^3`
SCERT Maharashtra solutions for Mathematics and Statistics (Commerce) [English] 12 Standard HSC 1.3 Differentiation Q.6
Activity: [4 Marks]
y = (6x4 – 5x3 + 2x + 3)6, find `("d"y)/("d"x)`
Solution: Given,
y = (6x4 – 5x3 + 2x + 3)6
Let u = `[6x^4 - 5x^3 + square + 3]`
∴ y = `"u"^square`
∴ `("d"y)/"du"` = 6u6–1
∴ `("d"y)/"du"` = 6( )5
and `"du"/("d"x) = 24x^3 - 15(square) + 2`
By chain rule,
`("d"y)/("d"x) = ("d"y)/square xx square/("d"x)`
∴ `("d"y)/("d"x) = 6(6x^4 - 5x^3 + 2x + 3)^square xx (24x^3 - 15x^2 + square)`
The rate of change of demand (x) of a commodity with respect to its price (y), if y = 20 + 15x + x3.
Solution: Let y = 20 + 15x + x3
Diff. w.r.to x, we get
`("d"y)/("d"x) = square + square + square`
∴ `("d"y)/("d"x)` = 15 + 3x2
∴ By derivative of the inverse function,
`("d"x)/("d"y) 1/square, ("d"y)/("d"x) ≠ 0`
∴ Rate of change of demand with respect to price = `1/(square + square)`
Find `("d"y)/("d"x)`, if y = x(x) + 20(x)
Solution: Let y = x(x) + 20(x)
Let u = `x^square` and v = `square^x`
∴ y = u + v
Diff. w.r.to x, we get
`("d"y)/("d"x) = square/("d"x) + "dv"/square` .....(i)
Now, u = xx
Taking log on both sides, we get
log u = x × log x
Diff. w.r.to x,
`1/"u"*"du"/("d"x) = x xx 1/square + log x xx square`
∴ `"du"/("d"x)` = u(1 + log x)
∴ `"du"/("d"x) = x^x (1 + square)` .....(ii)
Now, v = 20x
Diff.w.r.to x, we get
`"dv"/("d"x") = 20^square*log(20)` .....(iii)
Substituting equations (ii) and (iii) in equation (i), we get
`("d"y)/("d"x)` = xx(1 + log x) + 20x.log(20)
Find `("d"y)/("d"x)`, if x = em, y = `"e"^(sqrt("m"))`
Solution: Given, x = em and y = `"e"^(sqrt("m"))`
Now, y = `"e"^(sqrt("m"))`
Diff.w.r.to m,
`("d"y)/"dm" = "e"^(sqrt("m"))("d"square)/"dm"`
∴ `("d"y)/"dm" = "e"^(sqrt("m"))*1/(2sqrt("m"))` .....(i)
Now, x = em
Diff.w.r.to m,
`("d"x)/"dm" = square` .....(ii)
Now, `("d"y)/("d"x) = (("d"y)/("d"m))/square`
∴ `("d"y)/("d"x) = (("e"sqrt("m"))/square)/("e"^"m")`
∴ `("d"y)/("d"x) = ("e"^(sqrt("m")))/(2sqrt("m")*"e"^("m")`
Solutions for 1.3: Differentiation
SCERT Maharashtra solutions for Mathematics and Statistics (Commerce) [English] 12 Standard HSC chapter 1.3 - Differentiation
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Concepts covered in Mathematics and Statistics (Commerce) [English] 12 Standard HSC chapter 1.3 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 SCERT Maharashtra Mathematics and Statistics (Commerce) [English] 12 Standard HSC solutions Differentiation exercise by students is an easy way to prepare for the exams, as they involve solutions arranged chapter-wise and also page-wise. The questions involved in SCERT Maharashtra Solutions are essential questions that can be asked in the final exam. Maximum Maharashtra State Board Mathematics and Statistics (Commerce) [English] 12 Standard HSC students prefer SCERT Maharashtra Textbook Solutions to score more in exams.
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