If sec (x2-y2)/(x2+y2) = ea, then dy/dx is equal to (1) y2/x2 (2) y/x (3) x/y (4) (x2-y2)/(x2+y2) Solution: Given sec (x2-y2)/(x2+y2) = ea (x2-y2)/(x2+y2) = sec-1 ea Using... View Article
If f(x) = 1/(1-x), then the derivative of the composite function f[f{f(x)}] is 1) 0 2) 1/2 3) 1 4) 2 Solution: Given f(x) = 1/(1-x) f(f(x) = = (x-1)/x f(f(f(x))) = f((x-1)/x) = = x/(x-x+1) = x... View Article
If f(x) = cos x cos 2x cos 4x cos 8x cos 16x, then f ‘(Ï€/4) is equal to 1) √2 2) 1/√2 3) 0 4) √3/2 Solution: f(x) = cos x cos 2x cos 4x cos 8x cos 16x Multiply and divide by 2 sin x f(x) = (2 sin x... View Article
If y = sec-1((x+1)/(x-1)) + sin-1((x-1)/(x+1)), then dy/dx is equal to (1) 1 (2) 0 (3) (x-1)/(x+1) (4) (x+1)/(x-1) Solution: y = sec-1((x+1)/(x-1)) + sin-1((x-1)/(x+1)) = cos-1((x-1)/(x+1)) +... View Article
For |x|<1, y = 1 + x + x2 +…..∞, then dy/dx – y is equal to (1) x/y (2) x2/y2 (3) x/y2 (4) xy2 Solution: Given y = 1 + x + x2 +.....∞ This is an infinite GP with a = 1 and r = x. Sum, S∞ =... View Article
If [latex]y = \sqrt{\sin x+\sqrt{\sin x+\sqrt{\sin x+….\infty }}}[/latex] , then (2y-1)dy/dx is equal to (1) sin x (2) cos x (3) -sin x (4) -cos x Solution: Given y = √(sin x+ √(sin x + √(sin x +....∞ Squaring both sides y2 = sin x +... View Article
A differentiable function f(x) is defined for all x>0, and satisfies f(x3) = 4x4, for all x> 0. The value of f’(8) is (1) 16/3 (2) 32/3 (3) 16√2/3 (4) 32√2/3 Solution: f(x3) = 4x4 Differentiate w.r.t.x f’(x3) 3x2 = 16x3 f’(x3) = 16x/3 To... View Article
If y = logn x, where logn means log log log…(repeated n times), then x log x log2 x log3 x …logn–1 x logn x dy/dx is equal to 1) log x 2) x 3) 1/logx 4) logn x Solution: y = logn x Given logn means log log log...n times Let n = 2 Then y = log2x = log... View Article
If r = [2φ + cos2 (2φ + π/4)]1/2, then what is the value of the derivative of dr/dφ at φ = π/4? (1) 2(1/Ï€+1)1/2 (2) 2(2/Ï€+1)2 (3) (2/Ï€+1)1/2 (4) 2(2/Ï€+1)1/2 Solution: Given r = [2φ + cos2 (2φ + Ï€/4)]1/2 dr/dφ = (1/2)[2φ +... View Article
If x = y√(1-y2), then dy/dx is equal to (1) x (2) √(1-y2)/(1+2y2) (3) 0 (4) √(1-y2)/(1-2y2) Solution: Given x = y√(1-y2) Differentiate w.r.t.x 1 = √(1-y2)(dy/dx) +... View Article
If [latex]y = frac{e^{x}+e^{-x}}{e^{x}-e^{x}}[/latex] , then dy/dx is equal to (1) sech2x (2) cosech2x (3) -sech2x (4) -cosech2x Solution: y = (ex + e-x)/(ex - e-x) Use quotient rule dy/dx = [(ex - e-x)(ex -... View Article
If sin y = x sin (a+y), then dy/dx is equal to (1) sin a/sin2(a+y) (2) sin2(a+y)/sin a (3) sin a sin2(a+y) (4) sin2(a-y)/sin a Solution: sin y = x sin (a+y) sin y/sin (a+y) = x... View Article
If y = (1+1/x)(1+2/x)(1+3/x) ….(1+n/x) and x ≠0, then dy/dx where x = -1 is (1) n! (2) (n-1)! (3) -1n(n-1)! (4) -1nn! Solution: y = (1+1/x)(1+2/x)(1+3/x) ….(1+n/x) dy/dx = (-1/x2)(1+2/x)(1+3/x) ….(1+n/x) +... View Article
Let f be a twice differentiable function such that g’(x) = -f(x) and f’(x) = g(x), h(x) = {f(x)}2 + {g(x)}2. If h(5) = 11, then h(10) is equal to (1) 22 (2) 11 (3) 0 (4) 20 Solution: Given g’(x) = -f(x) and f’(x) = g(x) h(x) = {f(x)}2 + {g(x)}2 Differentiate w.r.t x... View Article
If y = [latex]x^{x^{x…\infty }}[/latex], then dy/dx is equal to If y = , then dy/dx is equal to (1) yxy-1 (2) y2/x(1- y log x) (3) y/x(1+y log x) (4) none of these Solution: Given y = Here... View Article
If y = 5x x5, then dy/dx is equal to (1) 5x (x5 log 5 - 5x4) (2) x5 log 5 - 5x4 (3) x5 log 5 + 5x4 (4) 5x (x5 log 5 + 5x4) Solution: Given y = 5x x5 dy/dx = 5x log... View Article
If [latex]y = \cos^{-1}\left ( \frac{1-\log x}{1+\log x} \right )[/latex] then dy/dx at x = e is (1) -1/e (2) -1/2e (3) 1/2e (4) 1/e Solution: y = cos-1[(1-log x)/(1+log x)] dy/dx = Put x = e = [-1/(√1-0)]×-(1+1)/e)/22... View Article
If y is a function of x and log(x + y) = 2xy, then the value of y'(0) is 1) 1 2) –1 3) 2 4) 0 Solution: log(x + y) = 2xy When x = 0, log y = 0 => y = 1 Differentiate both sides w.r.t.x... View Article
If y = sin-1(x/2) + cos-1(x/2), then the value of dy/dx is (1) 1 (2) -1 (3) 0 (4) 2 Solution: y = sin-1(x/2) + cos-1(x/2) We know sin-1θ + cos-1θ = π/2 So y = π/2 dy/dx = 0 ( derivative... View Article
If f(x) = ex, g(x) = sin-1x and h(x) = f(g(x)), then h’(x)/h(x) is equal to (1) (2) 1/√(1-x2) (3) sin-1 x (4) 1/(1-x2) Solution: Given f(x) = ex g(x) = sin-1x h(x) = f(g(x)) h(x) = f(sin-1x) h(x) =... View Article