Given y=x^(cosx), what is dy/dx?

The original form of the given function cannot be
differentiated. Therefore, first, we'll have to take natural logarithms both
sides:

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

Now, we'll apply power property of
logarithms:

ln y = cos x*ln
x

We'll differentiate both sides, using the product rule to
the right side:

y'/y = (cos x)'*ln x + cos x*(ln
x)'

y'/y = -sin x*ln x + (cos
x)/x

y' = y*[-sin x*ln x + (cos
x)/x]

But y = [x^(cos x)]

y' =
[x^(cos x)]*[-sin x*ln x + (cos
x)/x]

Therefore, dy/dx = [x^(cos x)]*[-sin
x*ln x + (cos x)/x].