12.4 Derivatives  (Page 3/18)

We have:

Substitute $f(a+h)={(a+h)}^2-3(a+h)+5$ and $f(a)=a^2-3a+5.$

$\begin{array}{ll}{f}^{\prime }(a)=\underset{h\to 0}{\lim }\frac{f\left(a+h\right)-f\left(a\right)}{h}\hfill & \hfill \\ =\underset{h\to 0}{\lim }\frac{\frac{3+\left(a+h\right)}{2-\left(a+h\right)}-\left(\frac{3+a}{2-a}\right)}{h}\hfill & \text{Substitute }f(a+h)\text{ and }f(a)\hfill \\ =\underset{h\to 0}{\lim }\frac{(2-(a+h))(2-a)\left[\frac{3+\left(a+h\right)}{2-\left(a+h\right)}-\left(\frac{3+a}{2-a}\right)\right]}{(2-(a+h))(2-a)(h)}\hfill & \text{Multiply numerator and denominator by }(2-(a+h))(2-a)\hfill \\ =\underset{h\to 0}{\lim }\frac{\cancel{\left(2-\left(a+h\right)\right)}(2-a)\left(\frac{3+\left(a+h\right)}{\cancel{\left(2-\left(a+h\right)\right)}}\right)-(2-(a+h))\cancel{\left(2-a\right)}\left(\frac{3+a}{\cancel{2-a}}\right)}{\left(2-\left(a+h\right)\right)(2-a)(h)}\hfill & \text{Distribute}\hfill \\ =\underset{h\to 0}{\lim }\frac{6-3a+2a-a^2+2h-ah-6+3a+3h-2a+a^2+ah}{\left(2-\left(a+h\right)\right)\left(2-a\right)(h)}\hfill & \text{Multiply}\hfill \\ =\underset{h\to 0}{\lim }\frac{5\cancel{h}}{\left(2-\left(a+h\right)\right)\left(2-a\right)(\cancel{h})}\hfill & \text{Combine like terms}\hfill \\ =\underset{h\to 0}{\lim }\frac{5}{\left(2-\left(a+h\right)\right)\left(2-a\right)}\hfill & \text{Cancel like factors}\hfill \\ =\frac{5}{\left(2-\left(a+0\right)\right)\left(2-a\right)}=\frac{5}{\left(2-a\right)\left(2-a\right)}=\frac{5}{{\left(2-a\right)}^2}\hfill & \text{Evaluate the limit}\hfill \end{array}$

We have

Multiply the numerator and denominator by the conjugate: $\frac{4\sqrt{a+h}+4\sqrt{a}}{4\sqrt{a+h}+4\sqrt{a}}.$