We have: #y = frac(sin(x))(2 + cos(x))#
This function can be differentiated using the "quotient rule":
#Rightarrow y' = frac((2 + cos(x)) cdot frac(d)(dx)(sin(x)) - (sin(x)) cdot frac(d)(dx)(2 + cos(x)))((2 + cos(x))^(2))#
#Rightarrow y' = frac((2 + cos(x)) cdot cos(x) - sin(x) cdot (- sin(x)))((2 + cos(x))^(2))#
#Rightarrow y' = frac(2 cos(x) + cos^(2)(x) + sin^(2)(x))((2 + cos(x))^(2))#
One of the Pythagorean identities is #cos^(2)(x) + sin^(2)(x) = 1#.
We can rearrange it to get:
#Rightarrow sin^(2)(x) = 1 - cos^(2)(x)#
Let's apply this rearranged identity:
#Rightarrow y' = frac(2 cos(x) + cos^(2)(x) + 1 - cos^(2)(x))((2 + cos(x))^(2))#
#Rightarrow y' = frac(1 + 2 cos(x))((2 + cos(x))^(2))#
