Solution assignment 20 Product and Quotient rule

Differentiate:

$y=\displaystyle\frac{xe^x+x^2}{x^2+1}$

We have to use the quotient rule but also the product rule. In the following a quote ' means that the concerning function has to be differentiated.

We have (using the product rule):

$f'(x)=[(xe^x)'+2x]=1\cdot{e^x}+xe^x+2x=(1+x)e^x+2x$

and:

$g'(x)=2x$

Now we can apply the quotient rule:

$y'=\displaystyle\frac{(x^2+1)[(1+x)e^x+2x]-(xe^x+x^2)\cdot2x}{(x^2+1)^2}$

This expression can be written fancier but that is outside the scope of this topic.