How do you differentiate #f(x) = ln(sqrt(arcsin(e^(3x)) ) # using the chain rule?
The first step we should take is to simplify the function by using logarithm rules to remove the square root from the function.
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To differentiate ( f(x) = \ln(\sqrt{\arcsin(e^{3x})}) ) using the chain rule, you would follow these steps:

Identify the outer function ( u ) and the inner function ( v ). ( u = \ln(v) ) ( v = \sqrt{\arcsin(e^{3x})} )

Differentiate the outer function with respect to its variable. ( \frac{du}{dv} = \frac{1}{v} )

Differentiate the inner function with respect to its variable. ( \frac{dv}{dx} = \frac{1}{2\sqrt{\arcsin(e^{3x})}} \cdot \frac{1}{\sqrt{1  (e^{3x})^2}} \cdot \frac{d}{dx}(e^{3x}) )

Apply the chain rule by multiplying the derivatives obtained in steps 2 and 3. ( \frac{df}{dx} = \frac{du}{dv} \cdot \frac{dv}{dx} )

Simplify the expression obtained in step 4.
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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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