use the four - step definition of the derivative to find $f(x)$ if $f(x)=sqrt{-7x - 18}$. you may have to…

Explanation:

Step1: Find $f(x + h)$

Substitute $x+h$ into $f(x)$: $f(x + h)=\sqrt{-7(x + h)-18}=\sqrt{-7x-7h - 18}$

Step2: Calculate $f(x + h)-f(x)$

[ \begin{align*} f(x + h)-f(x)&=\sqrt{-7x-7h - 18}-\sqrt{-7x - 18}\ \end{align*} ]

Step3: Compute $\frac{f(x + h)-f(x)}{h}$

[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{\sqrt{-7x-7h - 18}-\sqrt{-7x - 18}}{h}\ &=\frac{(\sqrt{-7x-7h - 18}-\sqrt{-7x - 18})(\sqrt{-7x-7h - 18}+\sqrt{-7x - 18})}{h(\sqrt{-7x-7h - 18}+\sqrt{-7x - 18})}\ &=\frac{(-7x-7h - 18)-(-7x - 18)}{h(\sqrt{-7x-7h - 18}+\sqrt{-7x - 18})}\ &=\frac{-7x-7h - 18 + 7x+18}{h(\sqrt{-7x-7h - 18}+\sqrt{-7x - 18})}\ &=\frac{-7h}{h(\sqrt{-7x-7h - 18}+\sqrt{-7x - 18})}\ &=\frac{-7}{\sqrt{-7x-7h - 18}+\sqrt{-7x - 18}} \end{align*} ]

Step4: Find $f'(x)$

Take the limit as $h\rightarrow0$: [ \begin{align*} f'(x)&=\lim_{h\rightarrow0}\frac{f(x + h)-f(x)}{h}\ &=\lim_{h\rightarrow0}\frac{-7}{\sqrt{-7x-7h - 18}+\sqrt{-7x - 18}}\ &=\frac{-7}{\sqrt{-7x - 18}+\sqrt{-7x - 18}}\ &=\frac{-7}{2\sqrt{-7x - 18}} \end{align*} ]

Answer:

$f(x + h)=\sqrt{-7x-7h - 18}$ $f(x + h)-f(x)=\sqrt{-7x-7h - 18}-\sqrt{-7x - 18}$ $\frac{f(x + h)-f(x)}{h}=\frac{-7}{\sqrt{-7x-7h - 18}+\sqrt{-7x - 18}}$ $f'(x)=\frac{-7}{2\sqrt{-7x - 18}}$