Originally Posted by
Chris L T521 Using the definition of the difference quotient, we have $\displaystyle \frac{x+h+\displaystyle\frac{1}{x+h}-\left[x+\displaystyle\frac{1}{x}\right]}{h}=\frac{h+\displaystyle\frac{1}{x+h}-\frac{1}{x}}{h}$
The goal here is to get the h term in the denominator to disappear.
First, we need to combine the terms in the numerator.
The common denominator is $\displaystyle x(x+h)$
Thus, the difference quotient becomes $\displaystyle \frac{\displaystyle\frac{hx(x+h)}{x(x+h)}+\frac{x} {x(x+h)}-\frac{x+h}{x(x+h)}}{h}=\frac{\displaystyle\frac{hx ^2+h^2x-h}{x(x+h)}}{h}=\frac{h\left(x^2+hx-1\right)}{h\left[x(x+h)\right]}$ $\displaystyle =\color{red}\boxed{\frac{x^2+hx-1}{x(x+h)}}$
Does this make sense?
By the way, to add on to what I told you in the other thread, when simplifying the difference quotient, the objective is to get the h term in the denominator to disappear (i.e. cancel out with something in the numerator of the difference quotient).