How do you differentiate #f(x)=csc(lnx) # using the chain rule?
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To differentiate ( f(x) = \csc(\ln(x)) ) using the chain rule:

Start by identifying the outer function and the inner function. In this case, the outer function is ( \csc(u) ) and the inner function is ( u = \ln(x) ).

Compute the derivative of the outer function with respect to its input. The derivative of ( \csc(u) ) is ( \csc(u) \cot(u) ).

Compute the derivative of the inner function with respect to ( x ). The derivative of ( \ln(x) ) is ( \frac{1}{x} ).

Apply the chain rule, multiplying the derivative of the outer function by the derivative of the inner function.
Putting it all together:
[ \begin{align*} f'(x) &= \csc(\ln(x))\cot(\ln(x)) \cdot \frac{1}{x} \ &= \frac{\csc(\ln(x))\cot(\ln(x))}{x} \end{align*} ]
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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