Question

Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

`(x^2 cos (pi/4))/sin x`

Answer 1

Let f(x) = `(x^2 cos(π/4))/sin x`

= `(x^2/sin x) cos π/4 = 1/sqrt2 (x^2/sin x)`

`d/dx (u/v) = (uv - uv)/v^2`

∴ `f'(x) = 1/sqrt2 xx ((d/dx x^2)sin x -x^2 d/dx sin x)/sin^2 x`

= `1/sqrt2 xx (2x sin x - x^2 cos x)/(sin^2 x)`

= `(x(2 sin x - x cos x))/(sqrt2 sin^2 x)` or `(x cos  π/4(2 sin x - x cos x))/sin^2 x`