COMPACT QUESTIONS (DIFFERENTIATION)

EXERCISE -1
1) If y= sin⁻¹x + sin⁻¹√(1- x²), find the value of dy/dx.                                        0

2) If y= log₁₀x , find x dy/dx.           log₁₀e

3) Differentiate sin⁻¹x w.r.t.cos⁻¹√(1- x²).        1

4) dy/dx of tan⁻¹{2x/(1- x²)} w.r.t sin⁻¹ {2x/(1+ x²)}.           1

5) Find the differential coefficient of x¹⁰ w r t. log₁₀x.           10logₑ10 . x¹⁰

6) Find points on the curve y= x + 1/x where dy/dx =0.                     (1,2)&(-1,-2)

7) If y= tan⁻¹{sinx/(1- cosx)}, find dy/dx.             -1/2

8) find x⁴ w.r.t.x².                                 2x²


EXERCISE -2

1) The differential coefficient of log (tan x) is
A) Secx B) cot x C) 2sec2x D) 2cosec2x

2) If y= tan⁻¹{cosx/(1+ sinx)}, then dy/dx
A) -1/2 B) -1 C) 1 D) 1/2

3) If y= sec⁻¹[(x+1)/(x -1)] + sin⁻¹[(x-1)/(x +1)] then dy/dx is
A) 1 B) (x-1)/(x +1) C) 0 D) (x+1)/(x -1)

4) The derivative of sin(cos⁻¹x) w r.t. cos⁻¹x is
A) cosx B) - x C) x D) Sinx

5) The derivative of xˣ w r.t. x log x is
A) x log x B) xˣ C) x⁻ˣ D) none 

6) The value of d/dx (sin x°) at x= 60, is
A) 0 B) 1/2 C) √3/2  D) none 

7) IF y= x + x²/2 + x³/3 + x⁴/4 + ..... to ∞ then dy/dx equal
A) y B) eʸ C) e ⁻ʸ D) none 

8) derivative of f(logx) w.r.t.x, where f(x)= log x, is
A) x/logx B) (logx)/x C) 1/(x logx) D) none

9) dy/dx of tan⁻¹{cosx - sinx)/(sinx + cosx) is
A) -2 B) 0 C) 1  D) none 

10) dy/dx of f(x)= x |x| is
A) 2x B) -2x C) 2|x | D) none 

11) If f(x) be an even function, then f'(0) is equal to
A) -1 B) 0 C) 1  D) none

EXERCISE -3

1) If y= 2x tan⁻¹x - logₑ(1+ x²), find dy/dx.                                             2 tan⁻¹x     

2) If tan²y = (1+ cos2x)/(1- cos2x), Find dy/dx.                            ±1

3) If y= log(1+ sin2x) + 2log sec(π/4 - x), show that dy/dx =0.

4) If log(xy)= x²+ y², find dy/dx when x= a, y = a.                                -1

5) If x²+ y²= t - 1/t and x⁴ + y⁴ = t²+ 1/t², then show that x³y dy/dx = 1.

6) If xʸ = eˣ⁻ʸ , then show that dy/dx = log x/(1+ logx)² = log x/(log ex)².

7) If y= ₑsin⁻¹x and z= ₑcos⁻¹x , show that dy/dx = constant.

8) If y=tan⁻¹x   ⁿ₀₁₂₃ₙⁿ₁₂₃ₙⁿ⁻¹₀₁₂ₙⁿ ₁₂₃ⁿₙʸˣᵇ ᵃ⁺ᵇ⁺²ˣ ₓ ˢᶦⁿˣ ˣᶜᵒˢˣ ˢᶦⁿˣ ⁻¹ₙₙ ₙ ₙ₋₁ₑˣ ʸˣ 










Miscellaneous -

1) If y= f{f(x)}, f(0)=0 and f'(0)=2, find y'(0).                                     4

2) Define differential coefficient of f(x) at x = a.

3) Given y = x³- 8x +7 and x = f(t). If x = 3, where t= 0 and df/dt =2, find dy/dt when t= 0.                      38

4) Show that the derivative of an even function is always an odd function.

5) If f(x)= logₑ x, g(x)= x², h(x)= eˣ and F(x) = f{g(h(x))}, then dF/dx is 
A) 2 B) 2x C) x² D) 0

6) 


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