Question

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed nonzero constants and m and n are integers): \(\frac{x^2cos(\frac{\pi}{4})}{sin\,x}\)

Answer 1

Let f(x)= \(\frac{x^2cos(\frac{\pi}{4})}{sin\,x}\)
By the quotient rule,
f'(x)= \(cos(\frac{\pi}{4})\). \(\frac{d}{dx}\)[\(\frac{sin\,x\frac{d}{dx}(x^2)-x^2\frac{d}{dx}(sin\,x)}{sin^2x}\)]
=\(cos(\frac{\pi}{4})\).[\(\frac{sin\,x.2x-x^2cosx}{sin^2x}\)]
=xcos\(\frac{\pi}{4}\) . \(\frac{[2sin\,x-x\,cosx]}{sin^2x}\)