Chapter 1.4, Problem 33E

In Exercise $33-36$, apply the three- step method to compute $f\text{'}(x)$ for the given function. Follow the steps that we used in Example $6$. Make sure to simplify the difference quotient as much as possible before taking its limits.

$f(x)=x^2+1$

Example 6.

Computing a Derivative from the Limit Definition

Use limits to compute the derivative $f\text{'}(5)$ for the following functions.

$f(x)=15-x^2$

$f(x)=\frac{1}{2x-3}$

Solution

$\frac{f(5+h)-f(5)}{h}=\frac{[15-{(5+h)}^2]-(15-5^2)}{h}$

Step 1: $=\frac{15-(25+10h+h^2)-(15-25)}{h}$

Step 2: $=\frac{-10h-h^2}{h}=-10-h$.

Therefore,

Step 3: $f\text{'}(5)=\underset{h\to 0}{\lim }(-10-h)=-10$.

$\frac{f(5+h)-f(5)}{h}=\frac{\frac{1}{2(5+h)-3}-\frac{1}{2(5)-3}}{h}$

Step 1: $=\frac{\frac{1}{7+2h}-\frac{1}{7}}{h}=\frac{\frac{7-(7+2h)}{(7+2h)7}}{h}$

Step 2: $=\frac{-2h}{(7+2h)7\times h}=\frac{-2}{(7+2h)7}=\frac{-2}{49+14h}$.

Therefore,

Step 3: $f\text{'}(5)=\underset{h\to 0}{\lim }\frac{-2}{49+14h}=-\frac{2}{49}$.