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While doing some work, I came across a problem in my high school textbook that asked me to differentiate

$$y = (2x+1)^2,$$

which I know how to do using chain rule. But the problem is that this problem was in the first section in the chapter of derivatives -- the chain rule isn't taught until later. Which leaves me wondering if there's another, simpler method of deriving this problem. I feel like it's really simple, but I can't figure it out.

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I assume you are asking to differentiate the function $y(x) = (2x+1)^2$. Perhaps the simplest way to differentiate $(2x+1)^2$ is to multiply it out and differentiate each summand.

Here, since $(2x+1)^2 = 4x^2 + 4x + 1$, you can see the derivative is $8x + 4$ without using the chain rule at all.


On a related note, the verb for "to take a derivative" is "to differentiate" and not "to derive."

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Using the product rule: $$y= (2x+1)^2$$ $$d/dx[(2x+1)(2x+1)]$$ $$d/dx(2x+1)(2x+1)+d/dx(2x+1)(2x+1)$$ $$2(2x+1)+2(2x+1)$$ $$8x+4$$

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