derivative question?

Question

what is the derivative of the following function:

{1-ln(cosx)+ln(tan[1-cos(x^2)]) }/{ 1+4sqrt(1+ln(cos x)+sqrt(x)) }

Answer 1

I'm not sure what the "..." is, but here are the steps:

let f(x) = g(x)/h(x), whcih I read as
g(x) = 1-ln(cosx) + ln(tan(1-cos(x^2)))
h(x) = 4+4sqrt(1+ln(cosx)+sqrt(x))

Then f'(x) = g'(x)/h(x) + -1*g(x)/h(x)^2

Now, you just need to be able to differentiate g(x) and h(x).
g'(x) is simply the sum of the derivatives of the three terms, the first of which drops out.
since the derivative of lnx is 1/x,
g'(x) = 1/cosx*(-sinx) (using the chain rule) Plus the derivative of the third term in g, which you can use the chain rule again to solve. Same for h(x)!

it's mostly an exercise in BEING CAREFUL not to lose any terms!