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): \(\frac{a+b\,sin\,x}{c+d\,cos\,x}\)

Answer 1

Let f (x)= \(\frac{a+b\,sin\,x}{c+d\,cos\,x}\)
By the quotient rule,
f'(x)= \(\frac{(c+d cos x)\frac{d}{dx}(a+bsin x)-(a+bsin x)\frac{d}{dx}(c+d cos x)}{(c+d cos x)^2}\)
=\(\frac{(c+d cos x)(b\,cos\,x)-(a+bsin x)(-d\,sin\,x)}{(c+d cos x)^2}\)
=\(\frac{cb\,cos\,x+bd\,cos^2\,x+ad\,sin\,x+bd\,sin^2\,x}{(c+d cos x)^2}\)
=\(\frac{bc\,cos\,x+ad\,sin\,x+bd(cos^2x+sin^2x)}{(c+d cos x)^2}\)
=\(\frac{bc\,cos\,x+ad\,sin\,x+bd}{(c+d cos x)^2}\)