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.