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Simplifying the difference quotient

Consider the function f(x)=x2+2xf(x) = x^2 + 2x, which represents the position of an object or the height of a curve depending on context.

For f(x)=x2+2xf(x) = x^2 + 2x, form the difference quotient f(x+h)−f(x)h\dfrac{f(x+h) - f(x)}{h} and simplify it as much as possible (the result should contain no hh in the denominator). Express your simplified answer in terms of xx and hh.

Hint

Start by computing f(x+h)f(x+h) by substituting (x+h)(x+h) everywhere you see xx in the formula for ff. Then subtract f(x)f(x), combine like terms, and factor out an hh from the numerator so it cancels with the hh in the denominator.

Step-by-step solution

First compute f(x+h)=(x+h)2+2(x+h)=x2+2xh+h2+2x+2hf(x+h) = (x+h)^2 + 2(x+h) = x^2 + 2xh + h^2 + 2x + 2h. Then f(x+h)−f(x)=(x2+2xh+h2+2x+2h)−(x2+2x)f(x+h) - f(x) = (x^2 + 2xh + h^2 + 2x + 2h) - (x^2 + 2x), which simplifies to 2xh+h2+2h2xh + h^2 + 2h after the x2x^2 and 2x2x terms cancel.

Now divide by hh: 2xh+h2+2hh=h(2x+h+2)h\dfrac{2xh + h^2 + 2h}{h} = \dfrac{h(2x + h + 2)}{h}. Since h≠0h \neq 0 in the limiting process, the hh factors cancel, leaving the simplified difference quotient 2x+h+22x + h + 2.

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