Is #f(x)=e^x/cosx-e^x/sinx# increasing or decreasing at #x=pi/6#?
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To determine if ( f(x) = \frac{e^x}{\cos x} - \frac{e^x}{\sin x} ) is increasing or decreasing at ( x = \frac{\pi}{6} ), we need to evaluate the derivative of ( f(x) ) and then substitute ( x = \frac{\pi}{6} ).
The derivative of ( f(x) ) with respect to ( x ) is given by: [ f'(x) = \frac{e^x}{\cos x} + e^x \tan x - \frac{e^x}{\sin x} - e^x \cot x ]
Now, substitute ( x = \frac{\pi}{6} ) into ( f'(x) ) to find the value of the derivative at ( x = \frac{\pi}{6} ): [ f'\left(\frac{\pi}{6}\right) = \frac{e^{\frac{\pi}{6}}}{\cos\left(\frac{\pi}{6}\right)} + e^{\frac{\pi}{6}} \tan\left(\frac{\pi}{6}\right) - \frac{e^{\frac{\pi}{6}}}{\sin\left(\frac{\pi}{6}\right)} - e^{\frac{\pi}{6}} \cot\left(\frac{\pi}{6}\right) ]
After evaluating this expression, we can determine whether ( f(x) ) is increasing or decreasing at ( x = \frac{\pi}{6} ) by analyzing the sign of ( f'\left(\frac{\pi}{6}\right) ). If ( f'\left(\frac{\pi}{6}\right) > 0 ), then ( f(x) ) is increasing at ( x = \frac{\pi}{6} ), and if ( f'\left(\frac{\pi}{6}\right) < 0 ), then ( f(x) ) is decreasing at ( x = \frac{\pi}{6} ).
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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.
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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