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Mathematics Question on Limits and derivations

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): cos x1+sin x\frac{cos\,x}{1+sin\,x}

Answer

Let f(x)=cos x1+sin x\frac{cos\,x}{1+sin\,x}
By quotient rule,
f'(x) = lim⁡h→0\lim_{h\rightarrow 0} f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}
=(1+sin x)ddx(cos x)ddx(1+sin x)(1+sin x)2\frac{(1+sin\,x)\frac{d}{dx}(cos\,x)\frac{d}{dx}(1+sin\,x)}{(1+sin\,x)^2}
=(1+sin x)(−sin x)−(cos x)(cos x)(1+sin x)2\frac{(1+sin\,x)(-sin\,x)-(cos\,x)(cos\,x)}{(1+sin\,x)^2}
=−sin x−sin2x−cos2x(1+sin x)2\frac{-sin\,x-sin^2x-cos^2x}{(1+sin\,x)^2}
=sin x−(sin2x+cos2x)(1+sin x)2\frac{sin\,x-(sin^2x+cos^2x)}{(1+sin\,x)^2}
=−sin x−1(1+sin x)2\frac{-sin\,x-1}{(1+sin\,x)^2}
=−(1+sin x)(1+sin x)2\frac{-(1+sin\,x)}{(1+sin\,x)^2}
=−1(1+sin x)-\frac{1}{(1+sin\,x)}