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How do you find the second derivative of #f(x)=x^2 lnx# ?

Answer 1

Christopher Agresta

#f''(x)=3+2lnx#

#"differentiate using the "color(blue)"product rule"#

#"given "f(x)=g(x)h(x)" then"#

#f'(x)=g(x)h'(x)+h(x)g'(x)larrcolor(blue)"product rule"#

#g(x)=x^2rArrg'(x)=2x#

#h(x)=lnxrArrh'(x)=1/x#

#f'(x)=x^2. 1/x+2xlnx=x+2xlnx#

#"differentiate "2xlnx" using the "color(blue)"product rule"#

#g(x)=2xrArrg'(x)=2#

#h(x)=lnxrArrh'(x)=1/x#

#d/dx(2xlnx)=2x . 1/x+2lnx=2+2lnx#

#f''(x)=1+2+2lnx=3+2lnx#

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Answer 2

Isaiah Allen

#f''(x)=2ln(x)+3#

By the product rule we get #f'(x)=2xln(x)+x^2*17x# simplifying #f'(x)=2xln(x)+x# #f''(x)=2ln(x)+2x*1/x+1# simplifying we get #f''(x)=2ln(x)+3#

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Answer 3

William Abdalla

To find the second derivative of ( f(x) = x^2 \ln(x) ), you first need to find the first derivative using the product rule, and then find the derivative of the resulting expression using the product rule again along with the chain rule.

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

Second derivative: [ f''(x) = \frac{d}{dx}\left(\frac{d}{dx}(x^2) \cdot \ln(x) + x^2 \cdot \frac{d}{dx}(\ln(x))\right) ]

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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.