Question 5: If y = x. log x then = …………..

Step-by-step solution provided.

Maharashtra State Board Class 12 (HSC Board) Commerce Mathematics & Statistics Solutions Chapter 3 Differentiation Miscellaneous Exercise 3 - Digest Answers | Maharashtra Board Solutions

Balbharati Maharashtra State Board12th Commerce Maths Solution Book PdfChapter 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 Maharashtra Board Solution

If y = (5x3 – 4x2 – 8x)9, then = ___________ (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)

Solution & Step-by-Step Answer:

(a) 9(5x3 – 4x2 – 8x)8 (15x2 – 8x – 8)

Question 2 Maharashtra Board Solution

If y = , then = ? (a) (b) (c) (d)

Solution & Step-by-Step Answer:

Question 3 Maharashtra Board Solution

If y = then = ? (a) (b) (c) 0 (d)

Solution & Step-by-Step Answer:

(a)

Question 4 Maharashtra Board Solution

If y = 2x2 + 22 + a2, then = ? (a) x (b) 4x (c) 2x (d) -2x

Solution & Step-by-Step Answer:

(b) 4x

Question 5 Maharashtra Board Solution

If y = 5x. x5, then = ? (a) 5x. x4(5 + log 5) (b) 5x. x5(5 + log 5) (c) 5x. x4(5 + x log 5) (d) 5x. x5(5 + x log 5)

Solution & Step-by-Step Answer:

Question 6 Maharashtra Board Solution

If y = then = ? (a) (b) (c) (d)

Solution & Step-by-Step Answer:

(b) Hint:

Question 7 Maharashtra Board Solution

If ax2 + 2hxy + by2 = 0, then = ? (a) (b) (c) (d)

Solution & Step-by-Step Answer:

(b)

Question 8 Maharashtra Board Solution

If x4. y5 = (x + y)(m+1) and then m = ? (a) 8 (b) 4 (c) 5 (d) 20

Solution & Step-by-Step Answer:

(a) 8 Hint: If xp. yq = (x + y)p+q, then ∴ m + 1 = 4 + 5 = 9 ∴ m = 8.

Question 9 Maharashtra Board Solution

If x = , y = then = ? (a) (b) (c) (d)

Solution & Step-by-Step Answer:

(d) Hint:

Question 10 Maharashtra Board Solution

If x = 2at2, y = 4at, then = ? (a) (b) (c) (d)

Solution & Step-by-Step Answer:

(c)

(II) Fill in the blanks:

Question 1 Maharashtra Board Solution

If 3x2y + 3xy2 = 0 then = …………

Solution & Step-by-Step Answer:

-1 Hint: 3x2y + 3xy2 = 0 ∴ 3xy(x + y) = 0 ∴ x + y = 0 ∴ y = -x ∴ = -1

Question 2 Maharashtra Board Solution

If xm. yn = (x+y)(m+n) then

Solution & Step-by-Step Answer:

Question 3 Maharashtra Board Solution

If 0 = log(xy) + a then

Solution & Step-by-Step Answer:

Question 4 Maharashtra Board Solution

If x = t log t and y = tt then = …………

Solution & Step-by-Step Answer:

y Hint: x = t log t = log tt = log y ∴ 1 = ∴ = y

Question 5 Maharashtra Board Solution

If y = x. log x then = …………..

Solution & Step-by-Step Answer:

Question 6 Maharashtra Board Solution

If y = [log(x)]2 then = …………..

Solution & Step-by-Step Answer:

Hint:

Question 7 Maharashtra Board Solution

If x = y + then = …………

Solution & Step-by-Step Answer:

Hint:

Question 8 Maharashtra Board Solution

If y = eax, then x. = …………

Solution & Step-by-Step Answer:

axy

Question 9 Maharashtra Board Solution

If x = t. log t, y = tt then = …………

Solution & Step-by-Step Answer:

Question 10 Maharashtra Board Solution

If y = then = ………

Solution & Step-by-Step Answer:

my Hint:

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

Question 1 Maharashtra Board Solution

If f’ is the derivative of f, then the derivative of the inverse of f is the inverse of f’.

Solution & Step-by-Step Answer:

False

Question 2 Maharashtra Board Solution

The derivative of loga x, where a is constant is .

Solution & Step-by-Step Answer:

True

Question 3 Maharashtra Board Solution

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

Solution & Step-by-Step Answer:

False

Question 4 Maharashtra Board Solution

The derivative of a polynomial is polynomial.

Solution & Step-by-Step Answer:

True

Question 5 Maharashtra Board Solution

Solution & Step-by-Step Answer:

False

Question 6 Maharashtra Board Solution

If y = log x, then .

Solution & Step-by-Step Answer:

True

Question 7 Maharashtra Board Solution

If y = e2, then = 2e.

Solution & Step-by-Step Answer:

False

Question 8 Maharashtra Board Solution

The derivative of ax is ax. log a.

Solution & Step-by-Step Answer:

True

Question 9 Maharashtra Board Solution

The derivative of xm. yn = (x + y)(m+n) is

Solution & Step-by-Step Answer:

False

(IV) Solve the following:

Question 1 Maharashtra Board Solution

If y = (6x3 – 3x2 – 9x)10, find

Solution & Step-by-Step Answer:

Given y = (6x3 – 3x2 – 9x)10

Question 2 Maharashtra Board Solution

If y = , find .

Solution & Step-by-Step Answer:

Question 3 Maharashtra Board Solution

If y = [log(log(log x))]2, find .

Solution & Step-by-Step Answer:

Question 4 Maharashtra Board Solution

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

Solution & Step-by-Step Answer:

Question 5 Maharashtra Board Solution

Find the rate of change of demand (x) of a commodity with respect to its price (y) if y =

Solution & Step-by-Step Answer:

Question 6 Maharashtra Board Solution

Find if y = xx.

Solution & Step-by-Step Answer:

y = xx ∴ log y = log xx = x log x Differentiating both sides w.r.t. x, we get

Question 7 Maharashtra Board Solution

Find if y = .

Solution & Step-by-Step Answer:

Question 8 Maharashtra Board Solution

Find , if y =

Solution & Step-by-Step Answer:

Question 9 Maharashtra Board Solution

Find if y = xx + (7x – 1)x

Solution & Step-by-Step Answer:

Question 10 Maharashtra Board Solution

If y = x3 + 3xy2 + 3x2y, find .

Solution & Step-by-Step Answer:

y = x3 + 3xy2 + 3x2y Differentiating both sides w.r.t. x, we get

Question 11 Maharashtra Board Solution

If x3 + y2 + xy = 7, find .

Solution & Step-by-Step Answer:

x3 + y2 + xy = 7 Differentiating both sides w.r.t. x, we get

Question 12 Maharashtra Board Solution

If x3y3 = x2 – y2, find .

Solution & Step-by-Step Answer:

x3y3 = x2 – y2 Differentiating both sides w.r.t. x, we get

Question 13 Maharashtra Board Solution

If x7. y9 = (x + y)16, then show that .

Solution & Step-by-Step Answer:

Question 14 Maharashtra Board Solution

If xa. yb = (x + y)a+b, then show that .

Solution & Step-by-Step Answer:

Question 15 Maharashtra Board Solution

Find if x = 5t2, y = 10t.

Solution & Step-by-Step Answer:

x = 5t2, y = 10t Differentiating x and y w.r.t. t, we get

Question 16 Maharashtra Board Solution

Find if x = e3t, y = .

Solution & Step-by-Step Answer:

x = e3t, y = Differentiating x and y w.r.t. t, we get

Question 17 Maharashtra Board Solution

Differentiate log(1 + x2) with respect to ax.

Solution & Step-by-Step Answer:

Let u = log(1 + x2) and v = ax Then we want to find Differentiating u and v w.r.t. x, we get

Question 18 Maharashtra Board Solution

Differentiate e(4x+5) with resepct to 104x.

Solution & Step-by-Step Answer:

Let u = e(4x+5) and v = 104x Then we want to find Differentiating u and v w.r.t. x, we get

Question 19 Maharashtra Board Solution

Find , if y = log x.

Solution & Step-by-Step Answer:

y = log x Differentiating w.r.t. x, we get Differentiating again w.r.t. x, we get

Question 20 Maharashtra Board Solution

Find , if y = 2at, x = at2.

Solution & Step-by-Step Answer:

x = at2, y = 2at Differentiating x and y w.r.t. t, we get

Question 21 Maharashtra Board Solution

Find , if y = x2. ex

Solution & Step-by-Step Answer:

y = x2. ex Differentiating w.r.t. x, we get = ex (2x + 2 + x2 + 2x) = ex (x2 + 4x + 2).

Question 22 Maharashtra Board Solution

If x2 + 6xy + y2 = 10, then show that .

Solution & Step-by-Step Answer:

x2 + 6xy + y2 = 10 ……..(1) Differentiating both sides w.r.t. a, we get

Question 23 Maharashtra Board Solution

If ax2 + 2hxy + by2 = 0, then show that = 0.

Solution & Step-by-Step Answer:

ax2 + 2hxy + by2 = 0 ……..(1) ∴ ax2 + hxy + hxy + by2 = 0 ∴ x(ax + hy) + y(hx + by) = 0 ∴ x(ax + hy) = -y(hx + by) ∴ …….(2) Differentiating (1) w.r.t. x, we get

Chapter 3 Differentiation Miscellaneous Exercise 3 Solutions