# Is #f(x)=cosx+cotx*sinx# increasing or decreasing at #x=pi/6#?

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To determine if ( f(x) = \cos(x) + \cot(x) \sin(x) ) is increasing or decreasing at ( x = \frac{\pi}{6} ), we need to find the derivative of ( f(x) ) and then evaluate it at ( x = \frac{\pi}{6} ).

The derivative of ( \cos(x) ) is ( -\sin(x) ), the derivative of ( \cot(x) ) is ( -\csc^2(x) ), and the derivative of ( \sin(x) ) is ( \cos(x) ).

Therefore, the derivative of ( f(x) ) is ( -\sin(x) - \cot(x)\csc^2(x) ).

Evaluating this derivative at ( x = \frac{\pi}{6} ), we get ( -\frac{\sqrt{3}}{2} - \frac{2\sqrt{3}}{3} = -\frac{5\sqrt{3}}{6} ), which is negative.

Since the derivative is negative at ( x = \frac{\pi}{6} ), the function is decreasing at that point.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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