3.4: The Product and Quotient Rules/ 3.5 Derivatives of Periodic Functions.

Main Points
3.4:
The derivative of the product is nto equal to the product of the derivatives! That is
f(x)*g(x) = x*x^2 = x^3 whose derivative is 3x^2 which is not equal to f '(x)*g'(x) = (1)(2x) = 2x. 3x^2 is not equal to 2x.
The product rule is stated as: u = f(x) and v = g(x); (fg)' = f 'g + fg'
also stated as d(uv)/dx = du/dx*v + dv/dx*u
The quotient rule is stated as: u = f(x) and v = g(x); (f/g)' = (f 'g - fg')/g^2
also stated as d/dx(u/v) = (du/dx*v - dv/dx * u)/v^2
3.5:
Derivatives of periodic functions are also periodic. d/dx(sin x) = cos x and d/dx(cos x) = -sin x

Challenges
The Liebniz notation is alot more confusing than the other. if both forms of the derivative function were not given I would have a lot more difficulty differentiating. Why would you want to multiply or divide derivatives?

Reflections
Is it possible that between sine and cosine, one or the other simply exists because it happens to be the derivative of the one of the two. ie. for example, cosine exists because it is the derivative of sine. Therefore cosine curve only exists because sine and its derivative were discovered first.