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): \(\frac{cos\,x}{1+sin\,x}\)

Answer 1

Let f(x)=\(\frac{cos\,x}{1+sin\,x}\)
By quotient rule,
f'(x) = \(\lim_{h\rightarrow 0}\) \(\frac{f(x+h)-f(x)}{h}\)
=\(\frac{(1+sin\,x)\frac{d}{dx}(cos\,x)\frac{d}{dx}(1+sin\,x)}{(1+sin\,x)^2}\)
=\(\frac{(1+sin\,x)(-sin\,x)-(cos\,x)(cos\,x)}{(1+sin\,x)^2}\)
=\(\frac{-sin\,x-sin^2x-cos^2x}{(1+sin\,x)^2}\)
=\(\frac{sin\,x-(sin^2x+cos^2x)}{(1+sin\,x)^2}\)
=\(\frac{-sin\,x-1}{(1+sin\,x)^2}\)
=\(\frac{-(1+sin\,x)}{(1+sin\,x)^2}\)
=\(-\frac{1}{(1+sin\,x)}\)