Differentiate the following w.r.t.x
(i) `3 x^(5//2)` (ii) `(4x)^(1//3)` (iii) `(6)/(sqrtx)` (iv) x sin nx

Answer

(i) `(15)/(2)x^(3//2)` (ii) `(4^(1//3))/(3)x^(-2//3)` (iii)` -3x^(-3//2)` (iv) `sin nx + nx cos nx`

Explanation

(i) `y =(d)/(dx)(3x^(5//2)) = 3(d)/(dx)(x^(5//2)) = 3xx(5)/(2)x^(3//2) = (15)/(2) x^(3//2)` (ii) `y= (d)/(dx)(4x)^(1)/(3) = (4x)^(1)/(3) = 4^(1)/(3)(d)/(dx)(x)^(1)/(3) = 4^(1)/(3) xx(1)/(3) x^(-2//3)`
(iii) `y = (d)/(dx)((6)/(sqrtx)) = 6(d)/(dx) (x ^(-1/2)/(2))= 6xx(- (1)/(2)x^(-3//2)) = -3x^(-3//2)`
(iv) `(d)/(dx) (x sin nx) = x(d)/(dx) (sin nx) +sin nx(d)/(dx)(x) = x(cos nx) n+ sin nx = nx cos nx +sin nx`