Maharashtra Board 12th Commerce Maths Solutions Chapter 3 Differentiation Miscellaneous Exercise 3

Balbharati Maharashtra State Board 12th Commerce Maths Solution Book Pdf Chapter 3 Differentiation Miscellaneous Exercise 3 Questions and Answers.

Maharashtra State Board 12th Commerce Maths Solutions Chapter 3 Differentiation Miscellaneous Exercise 3

(I) Choose the correct alternative:

Question 1.
If y = (5x³ – 4x² – 8x)⁹, then \(\frac{d y}{d x}\) = ___________
(a) 9(5x³ – 4x² – 8x)⁸ (15x² – 8x – 8)
(b) 9(5x³ – 4x² – 8x)⁹ (15x² – 8x – 8)
(c) 9(5x³ – 4x² – 8x)⁸ (5x² – 8x – 8)
(d) 9(5x³ – 4x² – 8x)⁹ (5x² – 8x – 8)
Answer:
(a) 9(5x³ – 4x² – 8x)⁸ (15x² – 8x – 8)

Question 2.
If y = \(\sqrt{x+\frac{1}{x}}\), then \(\frac{d y}{d x}\) = ?
(a) \(\frac{x^{2}-1}{2 x^{2} \sqrt{x^{2}+1}}\)
(b) \(\frac{1-x^{2}}{2 x^{2} \sqrt{x^{2}+1}}\)
(c) \(\frac{x^{2}-1}{2 x \sqrt{x} \sqrt{x^{2}+1}}\)
(d) \(\frac{1-x^{2}}{2 x \sqrt{x} \sqrt{x^{2}+1}}\)
Answer:
(c) \(\frac{x^{2}-1}{2 x \sqrt{x} \sqrt{x^{2}+1}}\)
Hint:

Question 3.
If y = \(e^{\log x}\) then \(\frac{d y}{d x}\) = ?
(a) \(\frac{e^{\log x}}{x}\)
(b) \(\frac{1}{x}\)
(c) 0
(d) \(\frac{1}{2}\)
Answer:
(a) \(\frac{e^{\log x}}{x}\)

Question 4.
If y = 2x² + 2² + a², then \(\frac{d y}{d x}\) = ?
(a) x
(b) 4x
(c) 2x
(d) -2x
Answer:
(b) 4x

Question 5.
If y = 5^x . x⁵, then \(\frac{d y}{d x}\) = ?
(a) 5^x . x⁴(5 + log 5)
(b) 5^x . x⁵(5 + log 5)
(c) 5^x . x⁴(5 + x log 5)
(d) 5^x . x⁵(5 + x log 5)
Answer:
(c) 5^x . x⁴(5 + x log 5)

Question 6.
If y = \(\log \left(\frac{e^{x}}{x^{2}}\right)\) then \(\frac{d y}{d x}\) = ?
(a) \(\frac{2-x}{x}\)
(b) \(\frac{x-2}{x}\)
(c) \(\frac{e-x}{ex}\)
(d) \(\frac{x-e}{ex}\)
Answer:
(b) \(\frac{x-2}{x}\)
Hint:

Question 7.
If ax² + 2hxy + by² = 0, then \(\frac{d y}{d x}\) = ?
(a) \(\frac{(a x+h y)}{(h x+b y)}\)
(b) \(\frac{-(a x+h y)}{(h x+b y)}\)
(c) \(\frac{(a x-h y)}{(h x+b y)}\)
(d) \(\frac{(2 a x+h y)}{(h x+3 b y)}\)
Answer:
(b) \(\frac{-(a x+h y)}{(h x+b y)}\)

Question 8.
If x⁴ . y⁵ = (x + y)^(m+1) and \(\frac{d y}{d x}=\frac{y}{x}\) then m = ?
(a) 8
(b) 4
(c) 5
(d) 20
Answer:
(a) 8
Hint:
If x^p . y^q = (x + y)^p+q, then \(\frac{d y}{d x}=\frac{y}{x}\)
∴ m + 1 = 4 + 5 = 9
∴ m = 8.

Question 9.
If x = \(\frac{e^{t}+e^{-t}}{2}\), y = \(\frac{e^{t}-e^{-t}}{2}\) then \(\frac{d y}{d x}\) = ?
(a) \(\frac{-y}{x}\)
(b) \(\frac{y}{x}\)
(c) \(\frac{-x}{y}\)
(d) \(\frac{x}{y}\)
Answer:
(d) \(\frac{x}{y}\)
Hint:

Question 10.
If x = 2at², y = 4at, then \(\frac{d y}{d x}\) = ?
(a) \(-\frac{1}{2 a t^{2}}\)
(b) \(\frac{1}{2 a t^{3}}\)
(c) \(\frac{1}{t}\)
(d) \(\frac{1}{4 a t^{3}}\)
Answer:
(c) \(\frac{1}{t}\)

(II) Fill in the blanks:

Question 1.
If 3x²y + 3xy² = 0 then \(\frac{d y}{d x}\) = …………
Answer:
-1
Hint:
3x²y + 3xy² = 0
∴ 3xy(x + y) = 0
∴ x + y = 0
∴ y = -x
∴ \(\frac{d y}{d x}\) = -1

Question 2.
If x^m . yⁿ = (x+y)^(m+n) then \(\frac{d y}{d x}=\frac{\ldots \ldots}{x}\)
Answer:
y

Question 3.
If 0 = log(xy) + a then \(\frac{d y}{d x}=\frac{-y}{\ldots . .}\)
Answer:
x

Question 4.
If x = t log t and y = t^t then \(\frac{d y}{d x}\) = …………
Answer:
y
Hint:
x = t log t = log t^t = log y
∴ 1 = \(\frac{1}{y} \cdot \frac{d y}{d x}\)
∴ \(\frac{d y}{d x}\) = y

Question 5.
If y = x . log x then \(\frac{d^{2} y}{d x^{2}}\) = …………..
Answer:
\(\frac{1}{x}\)

Question 6.
If y = [log(x)]² then \(\frac{d^{2} y}{d x^{2}}\) = …………..
Answer:
\(\frac{2(1-\log x)}{x^{2}}\)
Hint:

Question 7.
If x = y + \(\frac{1}{y}\) then \(\frac{d y}{d x}\) = …………
Answer:
\(\frac{y^{2}}{y^{2}-1}\)
Hint:

Question 8.
If y = e^ax, then x.\(\frac{d y}{d x}\) = …………
Answer:
axy

Question 9.
If x = t . log t, y = t^t then \(\frac{d y}{d x}\) = …………
Answer:
y

Question 10.
If y = \(\left(x+\sqrt{x^{2}-1}\right)^{m}\) then \(\sqrt{\left(x^{2}-1\right)} \frac{d y}{d x}\) = ………
Answer:
my
Hint:

(III) State whether each of the following is True or False:

Question 1.
If f’ is the derivative of f, then the derivative of the inverse of f is the inverse of f’.
Answer:
False

Question 2.
The derivative of log_a x, where a is constant is \(\frac{1}{x \cdot \log a}\).
Answer:
True

Question 3.
The derivative of f(x) = a^x, where a is constant is x . a^x-1
Answer:
False

Question 4.
The derivative of a polynomial is polynomial.
Answer:
True

Question 5.
\(\frac{d}{d x}\left(10^{x}\right)=x \cdot 10^{x-1}\)
Answer:
False

Question 6.
If y = log x, then \(\frac{d y}{d x}=\frac{1}{x}\).
Answer:
True

Question 7.
If y = e², then \(\frac{d y}{d x}\) = 2e.
Answer:
False

Question 8.
The derivative of a^x is a^x. log a.
Answer:
True

Question 9.
The derivative of x^m . yⁿ = (x + y)^(m+n) is \(\frac{x}{y}\)
Answer:
False

(IV) Solve the following:

Question 1.
If y = (6x³ – 3x² – 9x)¹⁰, find \(\frac{d y}{d x}\)
Solution:
Given y = (6x³ – 3x² – 9x)¹⁰

Question 2.
If y = \(\sqrt[5]{\left(3 x^{2}+8 x+5\right)^{4}}\), find \(\frac{d y}{d x}\).
Solution:

Question 3.
If y = [log(log(log x))]², find \(\frac{d y}{d x}\).
Solution:

Question 4.
Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = 25 + 30x – x².
Solution:

Question 5.
Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = \(\frac{5 x+7}{2 x-13}\)
Solution:

Question 6.
Find \(\frac{d y}{d x}\) if y = x^x.
Solution:
y = x^x
∴ log y = log x^x = x log x
Differentiating both sides w.r.t. x, we get

Question 7.
Find \(\frac{d y}{d x}\) if y = \(2^{x^{x}}\).
Solution:

Question 8.
Find \(\frac{d y}{d x}\), if y = \(\sqrt{\frac{(3 x-4)^{3}}{(x+1)^{4}(x+2)}}\)
Solution:

Question 9.
Find \(\frac{d y}{d x}\) if y = x^x + (7x – 1)^x
Solution:

Question 10.
If y = x³ + 3xy² + 3x²y, find \(\frac{d y}{d x}\).
Solution:
y = x³ + 3xy² + 3x²y
Differentiating both sides w.r.t. x, we get

Question 11.
If x³ + y² + xy = 7, find \(\frac{d y}{d x}\).
Solution:
x³ + y² + xy = 7
Differentiating both sides w.r.t. x, we get

Question 12.
If x³y³ = x² – y², find \(\frac{d y}{d x}\).
Solution:
x³y³ = x² – y²
Differentiating both sides w.r.t. x, we get

Question 13.
If x⁷ . y⁹ = (x + y)¹⁶, then show that \(\frac{d y}{d x}=\frac{y}{x}\).
Solution:

Question 14.
If x^a . y^b = (x + y)^a+b, then show that \(\frac{d y}{d x}=\frac{y}{x}\).
Solution:

Question 15.
Find \(\frac{d y}{d x}\) if x = 5t², y = 10t.
Solution:
x = 5t², y = 10t
Differentiating x and y w.r.t. t, we get

Question 16.
Find \(\frac{d y}{d x}\) if x = e^3t, y = \(e^{\sqrt{t}}\).
Solution:
x = e^3t, y = \(e^{\sqrt{t}}\)
Differentiating x and y w.r.t. t, we get

Question 17.
Differentiate log(1 + x²) with respect to a^x.
Solution:
Let u = log(1 + x²) and v = a^x
Then we want to find \(\frac{d u}{d v}\)
Differentiating u and v w.r.t. x, we get

Question 18.
Differentiate e^(4x+5) with resepct to 10^4x.
Solution:
Let u = e^(4x+5) and v = 10^4x
Then we want to find \(\frac{d u}{d v}\)
Differentiating u and v w.r.t. x, we get

Question 19.
Find \(\frac{d^{2} y}{d x^{2}}\), if y = log x.
Solution:
y = log x
Differentiating w.r.t. x, we get
\(\frac{d y}{d x}=\frac{d}{d x}(\log x)=\frac{1}{x}\)
Differentiating again w.r.t. x, we get
\(\frac{d^{2} y}{d x^{2}}=\frac{d}{d x}\left(\frac{1}{x}\right)=-\frac{1}{x^{2}}\)

Question 20.
Find \(\frac{d^{2} y}{d x^{2}}\), if y = 2at, x = at².
Solution:
x = at², y = 2at
Differentiating x and y w.r.t. t, we get

Question 21.
Find \(\frac{d^{2} y}{d x^{2}}\), if y = x² . e^x
Solution:
y = x² . e^x
Differentiating w.r.t. x, we get

= e^x (2x + 2 + x² + 2x)
= e^x (x² + 4x + 2).

Question 22.
If x² + 6xy + y² = 10, then show that \(\frac{d^{2} y}{d x^{2}}=\frac{80}{(3 x+y)^{3}}\).
Solution:
x² + 6xy + y² = 10 ……..(1)
Differentiating both sides w.r.t. a, we get

Question 23.
If ax² + 2hxy + by² = 0, then show that \(\frac{d^{2} y}{d x^{2}}\) = 0.
Solution:
ax² + 2hxy + by² = 0 ……..(1)
∴ ax² + hxy + hxy + by² = 0
∴ x(ax + hy) + y(hx + by) = 0
∴ x(ax + hy) = -y(hx + by)
∴ \(\frac{a x+h y}{h x+b y}=-\frac{y}{x}\) …….(2)
Differentiating (1) w.r.t. x, we get