Differentiation – JEE Main Practice Test

Let f : (0, ∞) → R be a function which is differentiable at all points of its domain and satisfies the condition x² f′(x) = 2x f(x) + 3, with f(1) = 4. Then 2f(2) is equal to:

If y = sec (tan⁻¹ x), then dy/dx at x = 1 is equal to:

The value of logₑ 2 · d/dx (log cos x · cosec x) at x = π/4 is:

If logₑ y = 3 sin⁻¹ x, then (1 − x²)y″ − xy′ at x = 1/2 is equal to:

If y² + logₑ (cos² x) = y, x ∈ (−π/2, π/2), then:

d²x/dy² equals:

If f(x) = | cos x   x   1 |       | 2 sin x   x²   2x |       | tan x   x   1 | then lim (x→0) f′(x)/x is:

If x logₑ (logₑ x) − x² + y² = 4 (y > 0), then dy/dx at x = e is equal to:

Let f¹(x) = (3x + 2)/(2x + 3), x ∈ R − {−3/2}. For n ≥ 2, define fⁿ(x) = f¹ ∘ fⁿ⁻¹(x). If f⁵(x) = (ax + b)/(bx + a), gcd(a, b) = 1, then a + b is equal to:

If y = y(x) is an implicit function of x such that logₑ(x + y) = 4xy, then d²y/dx² at x = 0 is equal to: