Problem using the difference quotient

Quote:

Originally Posted by astartleddeer

$\frac{f(x + \Delta x) - f(x)}{\Delta x} = \frac{\frac{7}{2}\left[(x +\Delta x)^\frac{1}{2} - x^\frac{1}{2}\right] + 3\left[ \frac{1}{(x + \Delta x)^2} - \frac{1}{x^2}\right]}{\Delta x}$

On the left side, rationalize the numerator (this is a strategy that is often helpful in situations like this), and on the right, combine fractions and expand.

$\lim_{\Delta x\to0}\frac{f(x+\Delta x) - f(x)}{\Delta x} = \lim_{\Delta x\to0}\frac{7(x+\Delta x)^{1/2}-7x^{1/2}}{2\Delta x} + \lim_{\Delta x\to0}\frac1{\Delta x}\left[\frac3{(x+\Delta x)^2} - \frac3{x^2}\right]$

$= \frac72\lim_{\Delta x\to0}\frac{(x+\Delta x)^{1/2}-x^{1/2}}{\Delta x} + 3\lim_{\Delta x\to0}\frac{x^2 - (x+\Delta x)^2}{x^2\Delta x(x+\Delta x)^2}$

$= \frac72\lim_{\Delta x\to0}\frac{\left[(x+\Delta x)^{1/2}-x^{1/2}\right]\left[(x+\Delta x)^{1/2}+x^{1/2}\right]}{\Delta x\left[(x+\Delta x)^{1/2}+x^{1/2}\right]} + 3\lim_{\Delta x\to0}\frac{x^2 - \left(x^2+2x\Delta x+\Delta x^2\right)}{x^2\Delta x(x+\Delta x)^2}$

$= \frac72\lim_{\Delta x\to0}\frac{(x+\Delta x)-x}{\Delta x\left[(x+\Delta x)^{1/2}+x^{1/2}\right]} - 3\lim_{\Delta x\to0}\frac{2x\Delta x+\Delta x^2}{x^2\Delta x(x+\Delta x)^2}$

$= \frac72\lim_{\Delta x\to0}\frac{\Delta x}{\Delta x\left[(x+\Delta x)^{1/2}+x^{1/2}\right]} - 3\lim_{\Delta x\to0}\frac{\Delta x\left(2x+\Delta x\right)}{x^2\Delta x(x+\Delta x)^2}$

$= \frac72\lim_{\Delta x\to0}\frac1{(x+\Delta x)^{1/2}+x^{1/2}} - 3\lim_{\Delta x\to0}\frac{2x+\Delta x}{x^2(x+\Delta x)^2}$

$= \frac7{2\left(x^{1/2}+x^{1/2}\right)} - \frac{3(2x)}{x^2\cdot x^2}$

$= \frac7{4\sqrt x} - \frac6{x^3}$