Question

Is there more than one way to differentiate `(2x+1)^2/(2x+4)`?

Answer 1

Yes, there is more than one way to differentiate. But there is only one derivative.

`f(x)=(2x+1)^2/(2x+4)=(4x^2+4x+1)/(2x+4)=(2x+1)^2(2x+4)^(-1)`

Written the first two ways, many would use the quotient rule. Written the last way it looks like a product rule problem. The first and last forms require the chain rule, but the middle form does not.

Just as the function `f` can be expressed in several ways, so too can the derivative `f'`.

`f'(x)=(4(2x+1)(2x+4)-2(2x+1)^2)/(2x+4)^2`

`=(2(2x+1)(2x+7))/(2x+4)^2=((2x+1)(2x+7))/(2(x+2)^2)`.

Of course, there are other expressions as well.