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How do you differentiate f(x) = x² ln(x) ?

Answer 1

Aurora Alvarado

Using the product rule,

#f^'(x) = x(2ln(x) + 1)#

We have that for every #f(x)# such that #f(x) = g(x)h(x)#, #f^'(x) = g^'(x)h(x) + g(x)h^'(x)#,

We know that #(x^2)^' = 2x# and that #(ln(x))^' = 1/x#, so we just evaluate it

#f^'(x) = 2x*ln(x) + x^2*1/x = 2x*ln(x) + x#

Or, putting #x# in evidence

#f^'(x) = x(2ln(x) + 1)#

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Answer 2

Scarlett Allison

To differentiate ( f(x) = x^2 \ln(x) ), you can use the product rule. The formula for the product rule is: ( (uv)' = u'v + uv' ), where ( u = x^2 ) and ( v = \ln(x) ). After differentiation, you get ( f'(x) = 2x\ln(x) + x ).

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Answer from HIX Tutor

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.