How do you differentiate #f(x)= x * (4-x^2)^(1/2) *ln x# using the product rule?
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To differentiate the function ( f(x) = x \cdot (4 - x^2)^{1/2} \cdot \ln(x) ) using the product rule, you would differentiate each term separately and then apply the product rule formula.
Let ( u = x ) and ( v = (4 - x^2)^{1/2} \cdot \ln(x) ).
Using the product rule: [ \frac{d}{dx} (u \cdot v) = u'v + uv' ]
Now, differentiate each term: [ u' = 1 ] [ v' = \frac{d}{dx} [(4 - x^2)^{1/2} \cdot \ln(x)] ]
To find ( v' ), you need to use the product rule again for ( (4 - x^2)^{1/2} ) and ( \ln(x) ).
Let ( a = (4 - x^2)^{1/2} ) and ( b = \ln(x) ).
[ a' = \frac{d}{dx} [(4 - x^2)^{1/2}] ] [ b' = \frac{d}{dx} [\ln(x)] ]
Now, differentiate each term: [ a' = \frac{-x}{(4 - x^2)^{1/2}} ] [ b' = \frac{1}{x} ]
Now apply the product rule to ( a ) and ( b ): [ v' = a'b + ab' ]
Finally, substitute the values of ( u' ), ( v' ), ( u ), and ( v ) into the product rule formula and simplify the expression.
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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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