How Do You Differentiate f(x,y) = e^(2x) * ln(x/y^2) with Respect to x?

[Genomics]

f(x,y)=e2x*ln(x/y^2)

find f(subscript-x)

Can anyone help me with this differentiation please? I know you must treat y as a constant and differentiate with respect to x.

 

[HallsofIvy]

Just do it step by step. Since it is a product of two functions, use the product rule:
$f_x= (e^{2x})_x (ln(x/y^2))+ (e^{2x})(ln(x/y^2))_x$

Now do $(e^{2x})_x$ alone: the derivative of $e^x$ is just $e^x$ so by the chain rule the derivative is $e^{2x}(2x)_x= 2e^{2x}$.

Next do $(ln(x/y^2))_x$ alone: the derivative of ln(x) is 1/x so by the chain rule the derivative is $(y^2/x)(x/y^2)_x= (y^2/x)(1/y^2)= 1/x$.

Actually, that last could be done more easily: since $ln(x/y^2)= ln(x)- 2ln(y)$, the derivative is just 1/x.

Now put them all together.

 

[Genomics]

Thanks very much I understand now