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The function f (x) = 1 – x³ – x⁵ is decreasing for

1) 1 ≤ x ≤ 5 2) x ≤ 1 3) x ≥ 1 4) All values of x Solution:... View Article

If y = cos(3 cos^-1 x), then d³y/dx³ is equal to

1) 24 2) 27 3) 3 – 12 x2 4) -24 Solution: We know cos 3x = 4 cos3x –... View Article

Suppose that f(x) is a differentiable function such that f ‘(x) is continuous f ‘(0) = 1 and f ” (0) does not exist. If g(x) = x f ‘ (x). Then,

1) g'(0) does not exist 2) g'(0) = 0 3) g'(0) = 1 4) g'(0) = 2 Solution: Given f... View Article

If y = sin^-1 (2x√(1-x²), -1/√2 ≤ x ≤ 1/√2, then dy/dx is equal to

(1) x/√(1-x2) (2) 1/√(1-x2) (3) 2/√(1-x2) (4) 2x/√(1-x2) Solution: y = sin-1 (2x√(1-x2) Put x = sin θ θ =... View Article

If y = [latex]x^{2}+\frac{1}{x^{2}+\frac{1}{x^{2}+\frac{1}{x^{2}+…\infty }}}[/latex] , then dy/dx is equal to

(1) 2xy/(2y-x2) (2) xy/(y+x2) (3) xy/(y-x2) (4) 2x/(2 + x2/y) Solution: y = \(\begin{array}{l}x^{2}+\frac{1}{x^{2}+\frac{1}{x^{2}+\frac{1}{x^{2}+…\infty }}}\end{array} \) y = x2 +... View Article

[latex]\frac{d}{dx}\sqrt{\frac{1-\sin 2x}{1+\sin 2x}}[/latex] is equal to

1) sec2x 2) -sec2(π/4 – x) 3) sec2(π/4 + x) 4) sec2(π/4 – x) Solution: Let y = √((1-sin 2x)/(1+sin... View Article

If x^m yⁿ = (x + y)^m+n, then (dy/dx)_{x = 1, y =2} is equal to

1) 1/2 2) 2 3) 2m/n 4) m/2n Solution: Given xm yn = (x + y)m+n Take log on both... View Article

If sin^-1 x + sin^-1y = π/2, then dy/dx is equal to

(1) x/y (2) -x/y (3) y/x (4) -y/x Solution: Given sin-1 x + sin-1y = π/2 sin-1 x = π/2... View Article

If sec (x²-y²)/(x²+y²) = e^a, 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 componendo... 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) = \(\begin{array}{l}\frac{1}{1-\frac{1}{1-x}}\end{array} \) = (x-1)/x f(f(f(x)))... View Article

If f(x) = cos x cos 2x cos 4x cos 8x cos 16x, then f ‘(π/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... 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)) + sin-1((x-1)/(x+1)) = π/2... 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... 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... 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) =... View Article

If y = logⁿ x, where logⁿ means log log log…(repeated n times), then x log x log² x log³x …log^n–1x 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... View Article

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