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}
=$\frac{({sin\,x-cos\,x})({sin\,x-cos\,x})-({sin\,x+cos\,x})({sin\,x+cos\,x})}{({sin\,x-cos\,x})^2}$
=$-\frac{({sin\,x-cos\,x})^2-({sin\,x+cos\,x})^2}{({sin\,x-cos\,x})^2}$
=$-\frac{[sin^2x+cos^2x–2sin xcos.x+sin^2x+cos^2x+2sinxcosx]}{({sin\,x-cos\,x})^2}$
=$\frac{-[1+1]}{({sin\,x-cos\,x})^2}$
=$\frac{-2}{({sin\,x-cos\,x})^2}$