What is the second derivative of #f(x) = x^2ln x #?

Answer 1

#3+ 2lnx#

Let's get the first derivative f'(x) first by applying the product rule: #f(x) = x^2lnx# #f'(x) = x^2(1/x) + (lnx)(2x)# #f'(x) = x + 2xlnx#

Get the second derivative f''(x) by differentiating the first derivative: Differentiate term by term. Apply product rule on the second term #f'(x) = x + 2xlnx# #f''(x) = 1 + 2x(1/x) + (lnx)(2)# #f''(x) = 1 + 2 + 2lnx# #f''(x) = 3+ 2lnx#

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Answer 2

Emily Ammerman

To find the second derivative of ( f(x) = x^2 \ln(x) ), first, compute the first derivative of ( f(x) ), then differentiate the result again with respect to ( x ).

First derivative: [ f'(x) = 2x \ln(x) + \frac{x^2}{x} = 2x \ln(x) + x ]

Second derivative: [ f''(x) = \frac{d}{dx}(2x \ln(x) + x) = 2 \ln(x) + 2 + 1 = 2 \ln(x) + 3 ]

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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.