Question

Differentiate w.r.t x y = sin x/ ( 1 + cos x)

Answer 1

derivation:
d/dx((sin(x))/(1+cos(x)))
Use the quotient rule, d/dx(u/v) = (v ( du)/( dx)-u ( dv)/( dx))/v^2, where u = sin(x) and v = cos(x)+1:
= ((1+cos(x)) (d/dx(sin(x)))-(d/dx(1+cos(x))) sin(x))/(1+cos(x))^2
The derivative of sin(x) is cos(x):
= (-((d/dx(1+cos(x))) sin(x))+(1+cos(x)) cos(x))/(1+cos(x))^2
Differentiate the sum term by term:
= (cos(x) (1+cos(x))-d/dx(1)+d/dx(cos(x)) sin(x))/(1+cos(x))^2
The derivative of 1 is zero:
= (cos(x) (1+cos(x))-sin(x) (d/dx(cos(x))+0))/(1+cos(x))^2
Simplify the expression:
= (cos(x) (1+cos(x))-(d/dx(cos(x))) sin(x))/(1+cos(x))^2
The derivative of cos(x) is -sin(x):
= (cos(x) (1+cos(x))--sin(x) sin(x))/(1+cos(x))^2
Simplify the expression:
(cos(x) (1+cos(x))+sin^2(x))/(1+cos(x))^2

=1/(1+cos x)