Find the derivative of the following functions with respect to x: (2+2)
(i) (x⁻¹ᐟ² - x¹ᐟ²)/(x⁻¹ᐟ² + x¹ᐟ²)
(ii) 3x/√(5+2x²)
For function (i), let (f(x) = \frac{x^{-1/2} - x^{1/2}}{x^{-1/2} + x^{1/2}}). Using the quotient rule, ((\frac{u}{v})' = \frac{u'v - uv'}{v^2}), ______ ___ ____ ______ _____.
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Solve the following differential equation:
(1/y²) (dy/dt) = 1 - e⁻³ᵗ
(c) Show that the equation
(y - 2x³)dx = x(1 - xy)dy
becomes exact on multiplication by x⁻² and solve it.
(dy/dx)² = (7x)/(4y²)
(a) Find the differential equation whose solution is given by
y = eˣ(A cos x + B sin x)
where, A and B are arbitrary constants.
(a) Evaluate the following limit.
(b) If the law of motion of a particle is given as: s = -t³ + 3t² + 25, then (2+1)
i) find its velocity and acceleration.
ii) find the distance covered by the particle in time t = 5 units
(a) Show that the set of following vectors form the sides of a right-angled triangle.
2î - ĵ + k̂
î - 3ĵ - 5k̂
3î - 4ĵ - 4k̂
Prove, with the help of vectors, that the diagonals of a parallelogram bisect each other.
Find A⁻¹, where
