DIFFERENTIATION
1) y= eˣ tanx + x logₑx. eˣ(tanx + sec²x) + (log x -1)
2) If y= (logx)/x + eˣ sin2x + log₅x, . (1- logx)/x² + eˣ(sin2x + cos2x) + 1/(logₑ5)
3) If x= exp[tan⁻¹{(y - x²)/x²}].
a) x[(1+ tan(logx)+ sec²x]
b) 2x[(1+ tan(logx)+ sec²x]
c) 2x[(1+ tan(logx)]+ secx]
d) 2x + x[(1+ tan(logx)]²
4) y= logₑ{tan⁻¹ √(1+ x²)}.
5) y= (x +1)(x +2)(x +3)
6) y= e⁵ˣ tan(x²+ 2).
7) y= (sinx)ˡᵒᵍˣ.
8) y= (x¹⁾² (1- 2x)²⁾³)/{(2- 3x)³⁾⁴(3- 4x)⁴⁾⁵.
9) y= xˣ.
10) y= ₑx. ₑx². ₑx³.ₑx⁴.
11) xʸ+ yˣ= 2.
12) y= sinx/(1+ cosx)/{1+ sinx/(1+ cosx...
13) x+ y = sin(x - y).
14) x²+ xeʸ + y =0, find y', also find the value of y' at point (0,0).
15) y= a cos t & x= a(t - sin t) find dy/dx at t=π/2. -1
16) Show that the function represented parametrically by the equations, x= (1+ t)/t³; y= 3/2t² + 2/t satisfies the relationship: x(y')³= 1+ y'.
17) find dy/dx at t=π/4 if y= cos⁴t & x= sin⁴t.
18) Find the slope of the tangent at a point P(t) on the curve x = at², y= 2at.
19) Differentiate logₑ(tanx) with respect to sin⁻¹(eˣ). e⁻ˣ√(1- e²ˣ)/(sinx cosx)
) If g is inverse of f and f'(x)= 1/(1+ xⁿ), then g'(c)=?
a) 1+ xⁿ b) 1+ [f(x)]ⁿ c) 1+ [g(x)]ⁿ d) none
) y= xˡᵒᵍˣ with respect to logx
) If g is inverse of f and f(x)= 2x + sinx; then g'(x) equas:
a) -3/x² + 1/√(1- x²)
b) 2+ sin⁻¹x c) 2+ cos g(x)
d) 1/(2+ cos g(x))
) If f(x)= x³+ x² f'(1)+ x f'(2)+ f'''(3) for all x ∈ R. Then find f(x) independent of f'(1), f'(2) and f'''(3).
) If x= a(t + sin t) and y= a(1- cos t) , find d²y/dx².
) y= f(x) and x= g(y) are inverse function of each other then express g'(y) and g"(y) in terms of derivative of f(x).
) If y = ₓₑx² then find y".
) Find y" at x=π/4, if y= x tan x.
) Show that y= eˣ sinx satisfies the relationship y" - 2y' + 2y = 0.
) (x x² x³
If f(x)= 1 2x 3x²
0 2 6x) find f'(x).
) If f(x)= |eˣ x²
log x sinx | find f'(1).
) 2x x² x³
If f(x)= x²+ 2x 1 3x +1
2x 1- 3x² 5x then find f'(1).
) d/dx of sin²[cot⁻¹√{(1+ x)/(1- x)}]=
a) -1/2 b) 0 c) 1/2 d) -1
) If f(x)= sin⁻¹{2x/(1+ x²)} then find
i) f'(2).
ii) f'(1/2)
iii) f'(1)
) If y= cos⁻¹(4x³- 3x). Then find
i) f'(-√3/2)
ii) f'(0)
iii) f'(√3/2)
) If √(1- x²)+ √(1- y²)= a(x - y), then show that dy/dx = √{(1- y²)/(1- x²)}
) Find second order derivative of y= sinx with respect to z= eˣ.
) If y= (tan⁻¹x)² then show (1+ x²)²+ 2x (1+ x²) y dy/dx = 2.
) Obtain differential coefficient of tan⁻¹{(√(1+ x²) - 1)/x} with respect to cos⁻¹√[{1+ √(1+ x²)}/2√(1+ x²)].
) y= (secx - tanx)/(secx - tanx) then dy/dx is
a) 2 secx(secx - tanx)
b) - 2 secx(secx - tanx)²
c) 2 secx(secx + tanx)²
d) - 2 secx(secx + tanx)²
) If y= (1+ x² + x⁴)/(1+ x + x²) and dy/dx = ax + b, then the values of a and b are
a) 2,1 b) -2,1 c) 2,-1 d) -2,-1
) Which of the following could be the sketch graph of y= d/dx (x log x)?
Let f(x)= x +3 log(x -2) & g(x)= x +5 log(x -1), then the set of x satisfying the inequality f'x)< g'(x) is
a) (2,7/2)
b) (1,2) U (7/2, ∞) c) (2, ∞) d) 7/2, ∞)
) Differential coefficient of (x^{(l+ m)/(m - n)})^1/(n - l) . (x^{(m+ n)/(n - l)})^1/(l - m). (x^{(n+ l)/(l - m)})^1/(m - n) w.r.t.x is
a) 1 b) 0 c) -1 d) xˡᵐⁿ
.
³²²²⁴ₑᶻˣˣlim ₓ→₀ˢᶦⁿˣⁿ lim ₓ→∞¹⁾ⁿ lim ₓ→₀ˣ²³²⁻¹⁻¹²⁻¹³²²²ˣ⁻¹²²²²²²⁻¹²⁻¹²²²⁴²∞∞∞ₓₓₓⁿᵐᵖᵐᵐⁿᵐᵖⁿᵖₑⁿᵖᵐⁿᵖᵐⁿᵖⁿᵖᵐ⁻¹²²²²¹⁰⁰θ⌈!*/-!#⊂⊂∩∪⨯≺∆∅-*𝔓∫Πₙ₌₁ⁿ⁽¹⁰¹⁻ⁿ⁾ˢᶦⁿˣ¹⁾²ₓ²ₓₓᵐⁿᵐ⁺ⁿ ²ʸˣʸˣʸˣˣʸʸˣˣʸʸˣₑʸ⁺∞θθθθθθ⁻¹⁻¹¹⁰²¹⁰²¹⁰¹⁰²¹⁰²² ₑ ∈
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