Chardonnay Felix

2021-08-18

Simplify the difference quotients between $f\left(x+h\right)-\frac{f\left(x\right)}{h}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}f\left(x\right)-\frac{f\left(a\right)}{x-a}$, if $f\left(x\right)=\sqrt{{x}^{2}-7}$

estenutC

$f\left(x\right)=\sqrt{{x}^{2}-7}$
$f\left(x+h\right)=\sqrt{{\left(x+h\right)}^{2}-7}$
$\frac{f\left(x+h\right)-f\left(x\right)}{h}=\frac{\sqrt{{\left(x+h\right)}^{2}-7}-\sqrt{{x}^{2}-7}}{h}$
$=\frac{\sqrt{{\left(x+h\right)}^{2}-7}-\sqrt{{x}^{2}-7}}{h}×\frac{\sqrt{{\left(x+h\right)}^{2}-7}-\sqrt{{x}^{2}-7}}{\sqrt{{\left(x+h\right)}^{2}-7}-\sqrt{{x}^{2}-7}}$
$=\frac{\left[{\left(x+h\right)}^{2}-7\right]-\left[{x}^{2}-7\right]}{h\left[\sqrt{{\left(x+h\right)}^{2}-7}+\sqrt{{x}^{2}-7}\right]}$
$=\frac{{\left(x+h\right)}^{2}-{x}^{2}}{h\left[\sqrt{{\left(x+h\right)}^{2}-7}+\sqrt{{x}^{2}-7}\right]}$
$=\frac{{h}^{2}+2xh}{h\left[\sqrt{{\left(x+h\right)}^{2}-7}+\sqrt{{x}^{2}-7}\right]}$
$=\frac{h+2x}{\sqrt{{\left(x+h\right)}^{2}-7}+\sqrt{{x}^{2}-7}}$
$f\left(x\right)=\sqrt{{x}^{2}-7},f\left(a\right)=\sqrt{{a}^{2}-7}$
$\frac{f\left(x\right)-f\left(a\right)}{x-a}=\frac{\sqrt{{x}^{2}-7}-\sqrt{{a}^{2}-7}}{x-a}$
$=\frac{\sqrt{{x}^{2}-7}-\sqrt{{a}^{2}-7}}{x-a}×\frac{\sqrt{{x}^{2}-7}-\sqrt{{a}^{2}-7}}{\sqrt{{x}^{2}-7}-\sqrt{{a}^{2}-7}}$

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