How do you differentiate #f(x)=xln(1/x)lnx/x #?
It is
The derivative is
#d(f(x))/dx=x'ln(1/x)+xln(1/x)'((lnx)'xx'lnx)/(x^2)=> d(f(x))/dx=ln(1/x)+x((1/x)'/(1/x))(1lnx)/x^2=> d(f(x))/dx=ln(1/x)1(1lnx)/x^2#
Remember that
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To differentiate the function ( f(x) = x\ln\left(\frac{1}{x}\right)  \frac{\ln x}{x} ), use the product rule and quotient rule as needed. Here are the steps:
 Apply the product rule to the term ( x\ln\left(\frac{1}{x}\right) ) and the quotient rule to the term ( \frac{\ln x}{x} ).
 Differentiate each term separately.
 Combine the derivatives to find the overall derivative of the function.
Let's differentiate step by step:

For the term ( x\ln\left(\frac{1}{x}\right) ): [ f_1(x) = x\ln\left(\frac{1}{x}\right) ] Apply the product rule: ( u = x ) and ( v = \ln\left(\frac{1}{x}\right) ). [ f_1'(x) = u'v + uv' ] [ f_1'(x) = (1)\ln\left(\frac{1}{x}\right) + x\left(\frac{1}{x^2}\right) ] [ f_1'(x) = \ln\left(\frac{1}{x}\right)  \frac{1}{x} ]

For the term ( \frac{\ln x}{x} ): [ f_2(x) = \frac{\ln x}{x} ] Apply the quotient rule: ( u = \ln x ) and ( v = x ). [ f_2'(x) = \frac{u'v  uv'}{v^2} ] [ f_2'(x) = \frac{(\frac{1}{x})(x)  (\ln x)(1)}{x^2} ] [ f_2'(x) = \frac{1}{x} + \frac{\ln x}{x^2} ]
Combine the derivatives:
[ f'(x) = f_1'(x) + f_2'(x) ] [ f'(x) = (\ln\left(\frac{1}{x}\right)  \frac{1}{x}) + (\frac{1}{x} + \frac{\ln x}{x^2}) ] [ f'(x) = \ln\left(\frac{1}{x}\right)  \frac{2}{x} + \frac{\ln x}{x^2} ]
So, the derivative of the function ( f(x) ) is:
[ f'(x) = \ln\left(\frac{1}{x}\right)  \frac{2}{x} + \frac{\ln x}{x^2} ]
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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