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

1 Answer
Jim H
Mar 22, 2015

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.

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