COMPACT QUESTIONS (DIFFERENTIATION 1ST ORDER)

EXERCISE -A

1) Find dy/dx of following:

a) log(sin eˣ).          (Cot eˣ)eˣ

b) {θ(x)}ⁿ.      n{θ(x)}ⁿ⁻¹ θ'(x)

c) log f(x).         f'(x)/f(x)

d) ₑf(x).                 ₑf(x). f'(x)

f) sin⁻¹x + sin⁻¹√(1- x²).            0

g) tan⁻¹{sin2x/(1- cosx)}.

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

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

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

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


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

b) tan⁻¹{2x/(1- x²)} w.r.t sin⁻¹{2x/(1+ x²)}.        1

c) x¹⁰ w.r.t log₁₀x.            10logₑ10. x¹⁰

d) x⁴ w.r.t x².               2x²

4) Prove the following:
a) If x³+y³-3axy=0, then dy/dx= (ay - x²)/(y²- ax). 


EXERCISE -B

1) The differential coefficient of log(tanx) is
a) sex b) cotx c) 2 sec2x d) 2 cosec2x

2) If y= tan⁻¹{cosx/(1+ sin2x)} , then dy/dx is
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)}, rhen 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) xˣ w.r.t. x log x is
a) xlogx  b) xˣ c) x⁻ˣ d) none

6) The value of d/dx (sinx°) at 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 is
a) y  b) eʸ c) ⁻ʸ d) none

8) The derivative of f(logx) w.r.t.x, where f(x)= logx, is
a) x/logx  b) logx/x c) 1/(xlogx) d) none

9) The derivative of tan⁻¹{(cosx - sinx)/(sinx + cosx).
a) -1 b) 0 c) 1  d) none

10) The derivative of the function, 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

12) If f(x)= logₑx, g(x)= x², h(x)= eˣ and F(x) = f{g{h(x)}}, then dF/dx is equal to
a) 2 b) 2x c) x² d) 0


EXERCISE -C

A) Find dy/dx
1) 2x tan⁻¹x - logₑ(1+ x²).           2tan⁻¹x

2) tan²y=(1+ cos2x)/(1- cos2c).          ±1

3) log√(x²+ y²)= tan⁻¹(y/x).              (x+y)/(x- y)

4) xʸ + yˣ = aᵇ.                      -(yxʸ⁻¹ + yˣlogy)/(xʸlogx+ xyˣ⁻¹)

5) y= xˢᶦⁿˣ + log sinxˣ.                   xˢᶦⁿˣ[sinx/x + cosx logx]+ cot(xˣ).xˣ(1+ logx)

6) y= sin⁻¹{x². √(1- x) - √x. √(1- x⁴)}.     2x/√(1- x⁴) - 1/{2√(x - x²)}

7) tan⁻¹{(tanx + sex -1)/(tanx - sex +1)}.        1/2

8) y= cot⁻¹{√(1+ t²) - t}; x=cot⁻¹{(1+ t)/(1- t)}.         -1/2

9) y= (sinx)ᶜᵒˢˣ + (cosx)ˢᶦⁿˣ.           1/(1+ x²)

10)  ₓsin⁻¹x with respect to sin⁻¹x.          ₓsin⁻¹x {logx + √(1- x²)/x . sin⁻¹x}

11)    ₑxˣ + √sin√x.                 ₑxˣ xˣ(1+ logx)+ cos√x/(4√(xsin√x))

12) xʸ yˣ= x+ y.          y/x[{x - y(x+y) - x(x+ y)logy}/{y(x+y) logx+ x(x+y)-y}] 

13) b tan⁻¹(x/a + tan⁻¹y/x).        b/a. (x²+ y²- ay)/{(x²+y²)sec²(y/b) - bx}

14)  ₓyˣ .               {ylogy(1+ xlogx logy)/(xlogx(1- xlogy))}

B) Show that:

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

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

3) If xʸ = eˣ⁻ʸ, then dy/dx= logx/(1+ logx)²= logx/(log ex)².

4) If y= ₑsin⁻¹x andₑcos⁻¹x, then dy/dz= constant.

5) If y= tan⁻¹{(a cosx - bsinx)/(bcosx + asinx)}, then dy/dx=-1

6) If y= cos⁻¹{(2 cosx +3sinx)/√13}, then dy/dx= 1.

7) If f(x)={(a+x)/(b+ x)}ᵃ⁺ᵇ⁺²ˣ, then f'(0)= {log(a²/b²) + (b²- a²)ab}(a/b)ᵃ⁺ᵇ.

8) If x= tan(y/2) + log tan(y/2) - 2 log(1+ tan(y/2)), then dy/dx= 1/2  siny (1+ cosy+ siny)

9) If y= cos⁻¹(8x⁴- 8x²+1), then dy/dx+ 4/√(1- x²)=0.

10) If y= m cos2θ(1- cos2θ) and x= K sin2θ(1+ cos2θ), then cotθ dy/dx = m/K.

11) If siny= x sin(a+ y), then dy/dx = {sin²(a+ y)}/sina = sina/(1- 2x cosa + x²).

12) If x - √(a²- y²)= a logx. y/{a+ √(a²- y²)} then dy/dx= y/√(a²- y²)

13) If y= log{(1+ x)/(1- x)}+ 1/2 log{(1+x +x²)/(1- x + x²)}+ √3 tan⁻¹{x √3/(1- x²)} then dy/dx= 6/(1- x⁶).

14) If y= x²/2 + x/2 √(x²+1) + log{√(x +√(x²+1))} then 2y= x dy/dx + log(dy/dx).

15) If √(1- x²) + √(1- y²)= a(x - y), then dy/dx= √{(1- y²)(1- x²)}.

16) {√(1+ x²) - √(1- x²}/{√(1+ x²) + √(1- x²)} w.r.t. √(1- x⁴) is {√(1- x⁴)-1}/x⁶

17) If logx = tan⁻¹{(y - x²)/x²}, then dy/dx= 2x{1+ tan(logx)}+ x sec²(logx).

C) Find the value

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

D) Differentiate

1) tan⁻¹[{√(1+ x²) -1}/x] w.r.t tan⁻¹x.        1/2


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