The quotient rule states that given functions u and v such that y = (u(x))/(v(x)), dy/dx = (u'(x)v(x) - u(x)v'(x))/(v^2(x)) By assigning u and v equal to the numerator and denominator, respectively, in the example function, we arrive at dy/dx = [(-sin x)(1+sin x) - (1+cos x)(cos x)]/(1+sin x)^2
When using the Quotient Rule, it is important to designate your functions u(x) and v(x) in such a way as to make things simple while still being accurate. In the above case, declaring cos(x) as u(x) would not allow us to use the quotient rule efficiently, as our numerator would then be 1 + u(x).
From the identities of trigonometric function derivatives, we know that the derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x). This could be proven using Euler's Formula, but for our purposes we shall accept these without proof. The derivative of any constant (such as 1 in our example) is 0, and the derivative of a sum is equal to the sum of the derivatives. Therefore:
u(x) = 1+cos(x), u'(x) = -sin(x), v(x) = 1+sin(x), v'(x) = cos(x)
dy/dx = [(-sin x)(1+sin x) - (1+cos x)(cos x)]/(1+sin x)^2
Attempting to use trig identities to simplify...
= [-sin(x) - sin^2(x) - cos(x) - cos^2(x)]/(1+sin x)^2 = [-1(sin^2(x)+cos^2(x)) -1 (sin x + cos x)]/(1+sin x)^2 = -[1 + sin(x) + cos(x)]/(1+sin(x))^2
However, the initial answer should be sufficient.
