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

Mar 22, 2015

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

$f \left(x\right) = {\left(2 x + 1\right)}^{2} / \left(2 x + 4\right) = \frac{4 {x}^{2} + 4 x + 1}{2 x + 4} = {\left(2 x + 1\right)}^{2} {\left(2 x + 4\right)}^{- 1}$

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 $f '$.

$f ' \left(x\right) = \frac{4 \left(2 x + 1\right) \left(2 x + 4\right) - 2 {\left(2 x + 1\right)}^{2}}{2 x + 4} ^ 2$

$= \frac{2 \left(2 x + 1\right) \left(2 x + 7\right)}{2 x + 4} ^ 2 = \frac{\left(2 x + 1\right) \left(2 x + 7\right)}{2 {\left(x + 2\right)}^{2}}$.

Of course, there are other expressions as well.