what is the derivative of the 1/sinx.cox?

Question

what is the derivative of the
1
---------
sinx.cox

please hurry :S

Answer 1

f (x) = cosec x sec x

f `(x)
(- cosec x cot x) sec x + (sec x tan x)(cosec x)
- cosec x sec x (cot x - tan x)

Answer 2

Do you mean, what's the derivative of 1 / (sin(x) cos(x))?

This is (sin(x) cos(x))^-1, so we have
-1*(sin(x) cos(x))^-2, times whatever the derivative of the inside is. The derivative of sin(x)cos(x), as with the product of any two functions, will be the deriviative of the first times the second, plus the first times derivative of the second. So it's: cos(x)cos(x) + sin(x)(-sin(x)), or cos^2(x) - sin^2(x),

Putting all of it together, you have
-1*(sin(x) cos(x))^-2 * (cos^2(x) - sin^2(x))

You can simplify this further though:

(sin^2(x) - cos^2(x)) / (sin(x) cos(x))^2
1/cos^2(x) - 1/sin^2(x)
sec^2(x) - csc^2(x)

Similarly, you could use the identity sin(2x) = 2sin(x)cos(x) to change the problem to something easier to derivate.