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How do you differentiate #e^x /lnx#?

Answer 1

Emma Aldridge

#(e^x(lnx-1/x))/(lnx)^2#

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

#"given "y=(g(x))/(h(x))" then"#

#dy/dx=(h(x)g'(x)-g(x)h'(x))/(h(x))^2larrcolor(blue)"quotient rule"#

#g(x)=e^xrArrg'(x)=e^x#

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

#rArrd/dx((e^x)/(lnx))#

#=(e^xlnx-1/x e^x)/(lnx)^2=(e^x(lnx-1/x))/(lnx)^2#

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Answer 2

William Abdalla

To differentiate ( \frac{e^x}{\ln x} ), use the quotient rule:

[ \frac{d}{dx} \left( \frac{e^x}{\ln x} \right) = \frac{\ln x \cdot \frac{d}{dx}(e^x) - e^x \cdot \frac{d}{dx}(\ln x)}{(\ln x)^2} ]

[ = \frac{\ln x \cdot e^x - e^x \cdot \frac{1}{x}}{(\ln x)^2} ]

[ = \frac{e^x (\ln x - \frac{1}{x})}{(\ln x)^2} ]

So, the derivative of ( \frac{e^x}{\ln x} ) is ( \frac{e^x (\ln x - \frac{1}{x})}{(\ln x)^2} ).

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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.