Maharashtra State Board Class 12 (HSC Board) Mathematics & Statistics Solutions Chapter 1 Differentiation Ex 1.5 - Digest Answers | Maharashtra Board Solutions

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Maharashtra State Board 12th Maths Solutions Chapter 1 Differentiation Ex 1.5

Question 1 Maharashtra Board Solution

Find the second order derivatives of the following: (i) 2x5 – 4x3 – – 9

Solution & Step-by-Step Answer:

Let y = 2x5 – 4x3 – – 9

(ii) e^2x. tan x
Solution:
Let y = e^2x. tan x

(iii) e^4x. cos 5x
Solution:
Let y = e^4x. cos 5x

(iv) x³. log x
Solution:
Let y = x³. log x

(v) log(log x)
Solution:
Let y = log(log x)

(vi) x^x
Solution:
y = x^x
log y = log x^x= x log x
Differentiating both sides w.r.t. x, we get

Question 2 Maharashtra Board Solution

Find of the following: (i) x = a(θ – sin θ), y = a (1 – cos θ)

Solution & Step-by-Step Answer:

x = a(θ – sin θ), y = a (1 – cos θ) Differentiating x and y w.r.t. θ, we get = a(1 – cos θ) …….(1)

(ii) x = 2at², y = 4at
Solution:
x = 2at², y = 4at
Differentiating x and y w.r.t. t, we get

(iii) x = sin θ, y = sin³θ at θ =
Solution:
x = sin θ, y = sin³θ
Differentiating x and y w.r.t. θ, we get,

(iv) x = a cos θ, y = b sin θ at θ =
Solution:
x = a cos θ, y = b sin θ
Differentiating x and y w.r.t. θ, we get

Question 3 Maharashtra Board Solution

(i) If x = at2 and y = 2at, then show that

Solution & Step-by-Step Answer:

x = at2, y = 2at ………(1) Differentiating x and y w.r.t. t, we get

(ii) If y = , show that
Solution:
y = ……..(1)

(iii) If x = cos t, y = e^mt, show that
Solution:
x = cos t, y = e^mt

(iv) If y = x + tan x, show that
Solution:
y = x + tan x

(v) If y = e^ax. sin (bx), show that y₂– 2ay₁+ (a²+ b²)y = 0.
Solution:
y = e^ax. sin (bx) ………(1)

(vi) If , show that
Solution:

(vii) If 2y = , show that 4(x²– 1)y₂+ 4xy₁– y = 0.
Solution:
2y = …… (1)
Differentiating both sides w.r.t. x, we get

(viii) If y = , show that
Solution:
y = =

(ix) If y = sin(m cos^-1x), then show that
Solution:
y = sin(m cos^-1x)
sin^-1y = m cos^-1x
Differentiating both sides w.r.t. x, we get

(x) If y = log(log 2x), show that xy₂+ y₁(1 + xy₁) = 0.
Solution:
y = log(log 2x)

(xi) If x²+ 6xy + y²= 10, show that
Solution:
x²+ 6xy + y²= 10 …… (1)
Differentiating both sides w.r.t. x, we get

(xii) If x = a sin t – b cos t, y = a cos t + b sin t, Show that
Solution:
x = a sin t – b cos t, y = a cos t + b sin t
Differentiating x and y w.r.t. t, we get

Question 4 Maharashtra Board Solution

Find the nth derivative of the following: (i) (ax + b)m

Solution & Step-by-Step Answer:

Let y = (ax + b)m

(ii)
Solution:
Let y =

(iii) e^ax+b
Solution:
Let y = e^ax+b

(iv) a^px+q
Solution:
Let y = a^px+q

(v) log(ax + b)
Solution:
Let y = log(ax + b)
Then

(vi) cos x
Solution:
Let y = cos x

(vii) sin(ax + b)
Solution:
Let y = sin(ax + b)

(viii) cos(3 – 2x)
Solution:

(ix) log(2x + 3)
Solution:

(x)
Solution:
Let y =

(xi) y = e^ax. cos (bx + c)
Solution:
y = e^ax. cos (bx + c)

(xii) y = e^8x. cos (6x + 7)
Solution:

Chapter 1 Differentiation Ex 1.5 Solutions