MULTIPLE CHOICE QUESTIONS (DIFFERENTIATION)

1) If ₑtan⁻¹{(y - x²)/x²} then dy/dx is
a) 2x[1+ tan(logx)]+ x sec²(logx)
b) x[1+ tan(logx)]+ sec²(logx)
c) 2x[1+ tan(logx)]+ x² sec²(logx)
d) 2x[1+ tan(logx)]+ sec²(logx)

2) If x= a cos⁴t , y= a sin⁴t, then dy/dx at t= 3π/4
a) 0 b) 1  c) -1 d) -2

3) d/dx(xˣ) is
a) xˣ(1- logx) b) xˣlogx c) xˣ⁺¹(1+ logx) d) xˣ(1+ logx) 

4) ₑx³ w.r.t. logx is
a) ₑx³ b) ₑx³. 3x³ c) ₑx³. 3x² d) ₑx³. 3x²+ 3x²

5) The second order derivative of a sin³t w r t. a cos³t at t=π/4
a) 2 b) 1/12a  c) 4√2/3a  b) 0

6) The derivative of the function f(x)= 3|x + y| at the point, x =-3 is
a) -3 b) 3 c) 0  d) does not exist

7) If y= √[x + √{x+ √(x +........∞ then the value of dy/dx is
a) x/(2y -1) b) 2/(2y -1) c) 1/(2y -1) d) x/(y -1)

8) If y= sinx + eˣ, then d²y/dx² is
a) (sinx - eˣ)/(cosx + eˣ)³
b)  1/(eˣ - sinx)
c)  (sinx - eˣ)/(cosx + eˣ)²
d)  (sinx + eˣ)/(cosx + eˣ)³

9) If sin⁻¹x+ sin⁻¹y=π/2, then the value of dy/dx is
a) x/y b) -x/y c) y/x d) -y/x

10) dx/dy= u and d²x/dy² = v, then the value of d²y/dx² is
a) - v/u² b) v/u²  c) - v/u³ d) v/u³

11) If y= √{sin√x}, then dy/dx is
a) 1/(2√{sin√x}) b) √{cos√x}/2x c) 1/(2√{cos√x}) d) cos√x/(4√x√{sin√x}

12) If f(x)= cos⁻¹{(1- (logx)²}/(1+ (logx)²)}, then f'(e) is
a) 2/e b) 1/e c) 1 d) 1/e²

13) If y= a cosmx - b sinmx, then d²y/dx² 
a) - m²y b) m²y c) -my d) my

14) If y= ₓeˣ, then dy/dx is
a) y(log x+ eˣ)
b) ylog x(1/2 + eˣ)
c) yeˣ(logx + 1/2)
d) yeˣ(x +logx)

15) If 2ˣ + 2ʸ = 2ˣ⁺ʸ, then dy/dx at x= y=1 is
a) 0 b) -1 c) 1 d) 2

16) If x= sin⁻¹t, y= log(2- t²), (0≤ t<1), then d²y/dx² at t=1/3 is
a) -9/4 b) -9/8 c) 9/4 d) 9/8

17) If y= (x + √(1+ x²))ⁿ, then (1+ x²)d²y/dx² + x dy/dx is
a) -y b) n²y c) - n²y d) 2n²y

18) If siny + e⁻ˣ = e, then the value of dy/dx at (1,π) is
a) 0 b) 1  c) e d) -1

19) If x= 2cost+ cos2t and y= 2sint - sin2t, then dy/dx at t=π/4 is
a) -(√2+1) b) √2 c) (√2-1) d) 1 - √2

20) If logx= z, then x² d²y/dx² is
a)  d²y/dz² b) d²y/dz²+ dy/dz c) d²y/dz²- dy/dz d) d²y/dz²-  2dy/dz

21) The value of x, at which the first derivative of the function (x + 1/x) w.r.t. x is 3/4
a) ±1/2 b) ±2/√3 c) ±√3/2 d) ±2

22) If f(x)= sin3x cos4x, then value of f'(π/2) is
a) 24 b) 25 c) -25 d) -24

23) The derivative of sec⁻¹{1/(2x²-1)} w r.t. √(1- x²) at x =1/2
a) 2 b) 4 c) 1 d) -2

24) The derivative of log₅(log₇x) (x >7) is
a) 1/(x logₑx) b) 1/(x logₑ5logₑ7)
c) 1/(x logₑ5logₑ7 logₑx) 
d) none

25) If 2y =(x - a)√(2ax - x²) + a² sin⁻¹{(x - a)/a}, then the value of dy/dx is
a) √(ax - x²) b) √(x²-ax) c) √(x² - 2ax) d) √(2ax - x²)

26) If y= x³ then the value of d²y/dx²/[1+ (dy/dx)²]³⁾² at the point (1,1) is
a) 3/5√10 b) 5/3√10 c) 4/3√10 d) 3/4√10 

27) If x= 1/z, y = f(x) and d²y/dx² = kz³ dy/dx + z⁴ d²y/dz², Then the value of k is
a) -1 b) 1 c) 2 d) -2

28) xy= ax²+ b/x, then the value of x²d²y/dx² + 2x dy/dx is
a) y b) -y  c) -2y  d) 2y

29) The derivative of the function (Secx + tanx)/(Secx - tanx) is
a) 2secx(Secx - tanx)  
b) 2sec²x(Secx + tanx)²
c) 2secx(Secx + tanx)²
d) 2secx(Secx + tanx)  

30) If y= sinx° and z= log₁₀x, then the value of dy/dx is
a) x° cosx°/log₁₀e
b) x° cosx°/logₑ10
c) x° cosx°/log₁₀e
d) x° cosx°/logₑ10

31) The value of d/dx [tan⁻¹{√x(3- x))/(1- 3x)] is
a) 3/{2(1- x)√x)}
b) 3/{2(1+ x)√x)}
c) 2/{(1+ x)√x)}
d) 3/{(1+ x)√x)}

32) If x= sint and y= cospt, then which of the following is true?
a) 

¹⁾ˣ ¹ ₓ ˣ ˣˣˣˣˣ ˣ ʸ ˣ⁺⁻¹ⁿ⁻ˣᶜᵒˢʸʸ ₐₑ⁻¹⁻¹⁻¹⁻¹⁻¹₅₇ₑₑₑₑₑ₇⁻¹³⁾² ᵗᵗⁿⁿ ₓₑ₂₁ˣ²ˣ⁻¹ₐₓₓₐₑ ₑᵐ⁺ᵐⁿˣ⁻ʸⁿ ˣʸ ⁻¹

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