If f(x)=((4sinx)/(2+cosx)), find f'(2)=?

Answer 1

You want the "quotient rule".

g, h, and f are all functions of x

if f = g/h, then f' = (g'h - gh') / h² ((How to remember this: "the top goes first"--ie. the first function to get differentiated is the one on top---and the bottom stays on the bottom)

So: f(x) = (4sinx) / (2+cosx)

f' = ( 4cosx * (2 + cosx) - 4sinx * -sinx ) / (2+cosx)²

rearrange and simplify the top:

8cosx + 4cos²x + 4sin²x cos² + sin² = 1, so:

8cosx + 4 and:

4 (1 + 2cosx )

The complete fraction is now: 4 ( 1 + 2cosx ) / ( 2 + cosx )²

To get the final answer for f(2) substitute 2 for x. I assume 2 is radians

For problems like this, it can be useful to use an online calculator to check your work

Answer 2

the derivative of f(x) is 4cos(x)/(2+cos(x)) +4sin²(x)/(2+cos(x))²

so f'(2) = .2674