Question

How do I find the derivative of `y= x arctan (2x) - (ln (1+4x^2))/4`?

Answer 1

You can use the Product Rule and Chain Rule to solve your problem.
1) Consider first `xarctan(2x)`; you have the product of `x` and `arctan(2x)` and the function of a function, `arctan(2x)` which requires the chain rule (derive the `arctan` and then multiply by the derivative of the argument `2x`).
You get:
`y'=1*arctan(2x)+x*1/(1+(2x)^2)*2`
2) Secondly you have the `-ln(1+4x^2)/4` bit. Here, again, you have the chain rule to deal with the `ln` of a function (the argument `1+4x^2`):
You get:
`y'=-1/4*1/(1+4x^2)*8x=-(2x)/(1+(2x)^2)`

Collecting the two parts you get:
`y'=arctan(2x)+(2x)/(1+(2x)^2)-(2x)/(1+(2x)^2)=arctan(2x)`

Hope it helps