# How do you differentiate #f(x)=x^2/ln(tanx)# using the quotient rule?

We use the quotient formula for finding derivatives

Let us use the formula

final answer should be

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To differentiate ( f(x) = \frac{x^2}{\ln(\tan(x))} ) using the quotient rule:

- Let ( u(x) = x^2 ) and ( v(x) = \ln(\tan(x)) ).
- Apply the quotient rule: ( f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} ).
- Find the derivatives:
- ( u'(x) = 2x ) (derivative of ( x^2 )).
- ( v'(x) = \frac{1}{\tan(x)} \cdot \sec^2(x) = \frac{\sec^2(x)}{\tan(x)} ) (derivative of ( \ln(\tan(x)) )).

- Substitute into the quotient rule formula: [ f'(x) = \frac{(2x) \cdot \ln(\tan(x)) - x^2 \cdot \frac{\sec^2(x)}{\tan(x)}}{(\ln(\tan(x)))^2} ]
- Simplify if needed.

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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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