# 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

(3)  Find the derivative of √(xy)  =  e x - y

(4)  Find the derivatives of the following

xy  =  yx

(5)  Find the derivatives of the following

y  =  (cos x) log x

(6)  Find the derivatives of the following

(x2/a2) + (y2/b2)  =  1

(7)  Find the derivatives of the following

√(x2 + y2)  =  tan-1(y/x)

(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)

(10)  Differentiate the following

tan-1[√(1 - cos x)/(1+cosx)]

(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)

(15)  Differentiate the following

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

(16)  Differentiate the following

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

(17)  Differentiate the following

sin-1(3x - 4x3)

(18)  Differentiate the following

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

(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

(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

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

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

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