Differentiate.

y= (cos x)^x

u= cos x
du= -sin x dx

ln y = ln(cos x)^x
ln y = x ln(cos x)

(dy/dx)/(y)= ln(cos x)
(dy/dx)= y ln(cos x)
= (cos x)^x * (ln cos x)

(dx/du)= x(cos x)^(x-1) * (-sin x)
= - x sin(x)cos^(x-1)(x)

(dy/dx)-(dx/du)= [(cos^x(x))(ln(cos(x)))-(x sin(x)cos^(x-1)(x)]

(dy/du)= cos^x(x)*(ln(cos(x)))-(x tan(x))

Is this correct?

Also, I am stuck on a different problem.

Differentiate.

y= arctan(arcsin(sqrt(x)))

u= arcsin(sqrt(x))
du= (1/(sqrt(1-x^2))) dx

ln y = ln ?? do I put the whole original here?

Your differentiation of y = (cos x)^x is almost correct. Here's the correct differentiation:

Start with:
y = (cos x)^x

First, let's take the natural logarithm (ln) of both sides:
ln y = ln((cos x)^x)

Then, we can use the logarithm rule that ln(a^b) = b * ln(a):
ln y = x * ln(cos x)

To differentiate both sides, we need to use the chain rule. On the left side, we have d(ln y)/dx, and on the right side, we have d(x * ln(cos x))/dx.

For the left side:
d(ln y)/dx = (1/y) * (dy/dx)

For the right side, we need to apply the product rule:
d(x * ln(cos x))/dx = x * d(ln(cos x))/dx + ln(cos x) * dx/dx

Now, let's differentiate each term separately:
d(ln(cos x))/dx = -sin x
dx/dx = 1

Substituting these results back into the equation, we have:
(1/y) * (dy/dx) = x * (-sin x) + ln(cos x)

Finally, we can multiply both sides by y and rearrange to solve for dy/dx:
dy/dx = y * [x * (-sin x) + ln(cos x)]
dy/dx = (cos x)^x * [x * (-sin x) + ln(cos x)]

Now, let's move on to your second question regarding differentiating y = arctan(arcsin(sqrt(x))).

First, you correctly identified u = arcsin(sqrt(x)), but the next step should be to differentiate u with respect to x.

Given:
u = arcsin(sqrt(x))
du/dx = (1/(sqrt(1 - x))) * (1/(2sqrt(x))) = 1/(2sqrt(x(1 - x)))

Since y = arctan(u), we can differentiate y using the chain rule:

dy/dx = (dy/du) * (du/dx)

For the first term, dy/du, we differentiate arctan(u) with respect to u:
(dy/du) = 1/(1 + u^2)

Then, we substitute the value of u and du/dx back in to get the final answer:
dy/dx = (1/(1 + (arcsin(sqrt(x)))^2)) * (1/(2sqrt(x(1 - x))))

For the last part of your question, to find the derivative of y = arctan(arcsin(sqrt(x))), there is no need to use the natural logarithm. The differentiation can be done directly using the chain rule as explained above.