PRACTICE PROBLEMS ON IMPLICIT DIFFERENTIATION

(1)  Find the derivative of y = x ^{cos x       Solution}

(2)  Find the derivative of y = x ^{log x  +} (log x)^x       Solution

(3)  Find the derivative of √(xy)  =  e ^{x - y}       Solution

(4)  Find the derivatives of the following

x^y  =  y^x        Solution

(5)  Find the derivatives of the following

y  =  (cos x) ^{log x}       Solution

(6)  Find the derivatives of the following

(x²/a²) + (y²/b²)  =  1        Solution

(7)  Find the derivatives of the following

√(x² + y²)  =  tan^-1(y/x)       Solution

(8)  Differentiate the following

tan (x + y) + tan (x - y) = x           Solution

(9)  Differentiate the following

If cos (xy) = x, show that dy/dx  =  -(1+ysin(xy))/x sin(xy)

          Solution

(10)  Differentiate the following

tan^-1[√(1 - cos x)/(1+cosx)]          Solution

(11)  Differentiate the following

tan^-1[6x/1-9x²]                Solution

(12)  Differentiate the following

cos (2tan^-1[√(1-x)/(1+x)])                Solution

(13)  Differentiate the following

x = a cos³t, y = a sin³t             Solution

(14)  Differentiate the following

x = a (cos t + t sin t) ; y = a (sin t - t cos t)           Solution

(15)  Differentiate the following

x = (1-t²)/(1+t²) ; y = 2t/(1+t²)           Solution

(16)  Differentiate the following

cos^-1(1 -x²)/(1+x²)            Solution

(17)  Differentiate the following

sin^-1(3x - 4x³)          Solution

(18)  Differentiate the following

tan^-1[(cos x + sin x) / (cos x - sin x)]          Solution

(19)  Find the derivative of sin x² with respect to x² 

Solution

(20)  Find the derivative of sin^-1(2x / (1 + x²)) with respect to tan^-1 x               Solution

(21)  Differentiate the following

If u = tan^-1 [√(1+x²) - 1]/x and v = tan^-1x, find du/dv    Solution

(22)  Find the derivative with tan^-1 (sin x/(1 + cos x)) with respect to tan^-1 (cos x/(1 + sin x))      Solution

(23)  If y = sin^-1x then find y''            Solution

(24)  If y = e^(tan^-1x) show that (1+ x² ) y'' + (2x −1) y' = 0.             Solution

(25)  If y = sin^-1 x/√(1-x²) show that (1-x²)y₂ - 3xy₁ - y = 0           Solution

(26)  If x = a (θ + sin  θ), y = a (1 - cos  θ) then prove that at  θ = π/2, y''  =  1/a          Solution

(27)  If sin y = x sin(a + y), then prove that dy/dx  =  sin² (a + y)/sin a , a ≠ nπ          Solution

(28)  If y = (cos^-1x)² prove that (1 -x²)(d²y/dx²) - x (dy/dx)  - 2 = 0          Solution

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