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): \(\frac{sin\,x+cos\,x}{sin\,x-cos\,x}\)

Answer 1

Let f(x)=\(\frac{sin\,x+cos\,x}{sin\,x-cos\,x}\)
By the quotient rule,
f'(x)= (\({sin\,x-cos\,x}\)) \(\frac{d}{dx}\) (\({sin\,x+cos\,x}\))–(\({sin\,x+cos\,x}\)) \(\frac{d}{dx}\)\(\frac{({sin\,x-cos\,x})}{({sin\,x-cos\,x})^2}\)
=\(\frac{({sin\,x-cos\,x})({sin\,x-cos\,x})-({sin\,x+cos\,x})({sin\,x+cos\,x})}{({sin\,x-cos\,x})^2}\)
=\(-\frac{({sin\,x-cos\,x})^2-({sin\,x+cos\,x})^2}{({sin\,x-cos\,x})^2}\)
=\(-\frac{[sin^2x+cos^2x–2sin xcos.x+sin^2x+cos^2x+2sinxcosx]}{({sin\,x-cos\,x})^2}\)
=\(\frac{-[1+1]}{({sin\,x-cos\,x})^2}\)
=\(\frac{-2}{({sin\,x-cos\,x})^2}\)