what is the derivative??? help!!!!?

Question

f(x) = sqrt(cos(x/(x+1)))
how do i find the derivative of that equation?

Answer 1

Basically, you use the chain rule a couple of times.

sqrt (cos (x / (x + 1)))
= [cos (x / (x + 1))]^(1/2)

derivative:

deriv of sqrt (____) * deriv of cos (___) * deriv of (x/x+1)

deriv of sqrt (___) = (1/2)[cos (x / x+1)]^(-1/2)
deriv of cos (___) = -sin (x / x+1)
deriv of x / x+1 = [(1)(x + 1) - (x)(1)] / (x + 1)^2
= 1 / (x + 1)^2

So, the derivative of the entire function is
(1/2)[cos (x / x+1)]^(-1/2) * [-sin (x / x+1)] * [1 / (x + 1)^2]

Answer 2

You would apply chain rule but rewriting it might help:

(cos(x/(x+1)))^-1/2

It should look something like this unless there is an error I have somehow missed:

-1/2(cos(x/(x+1)))^1/2 * sin(x/(x+1)) * 1/(x+1)^2

Answer 3

Use the chain rule.
dy/dx = dy/du * du/dx
where y is a function of u and u is a function of x.

Only what you've got going on here is actually a three-stage chain; so it's going to be
dy/dx = dy/dv * dv/du * du/dx
where y is a function of v, v is a function of u and u is a function of x.

u = x / x+1
v = cos u
y = v ** .5

Assuming you know how to differentiate each of those, the rest is easy.

Answer 4

{-.5{sin[x/(x+1)]}^-1/2} times
[1/(x+1)-x/(x+1)^2]