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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].
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