# How do you differentiate #f(x)=x(lnx)^3 # using the product rule?

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To differentiate ( f(x) = x(\ln x)^3 ) using the product rule, you would first identify two functions: ( u = x ) and ( v = (\ln x)^3 ). Then, apply the product rule: ( f'(x) = u'v + uv' ). The derivatives are ( u' = 1 ) and ( v' = 3(\ln x)^2 \cdot \frac{1}{x} ). Therefore, the derivative of ( f(x) ) is ( f'(x) = 1 \cdot (\ln x)^3 + x \cdot 3(\ln x)^2 \cdot \frac{1}{x} ). Simplifying gives ( f'(x) = (\ln x)^3 + 3(\ln x)^2 ).

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