91Ó°ÊÓ

Sketch a graph of the associated slope function ffor each functionf in Exercises 25–30.

Use (a) the $h→0$definition of the derivative and then

(b) the $z→c$definition of the derivative to find $f\text{'}\left(c\right)$ for each function f and value $x=c$ in Exercises 23–38.

29.$f\left(x\right)=x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},x=9$

Find the derivatives of each of the functions in Exercises 17–50. In some cases it may be convenient to do some preliminary algebra.

$f\left(x\right)=\frac{-2^x}{5x\sin x}.$

Fill in the blanks to complete each of the following theorem statements:

If a function $f$is differentiable at $x=c$, then $f$is at ---------$x=c$.

Fill in the blanks to complete each of the following theorem statements:

If $f$and $f^{-1}$are inverse functions, then for all appropriate values of $x$we can write the derivative of $f^{-1}\left(x\right)$in terms of the derivative of $f\left(x\right)$as ------------

Fill in the blanks to complete each of the following theorem statements:

$f^{\text{'}}\left(x\right)=kf\left(x\right)$for some constant $k$if and only if $f$is a/an ---------- function.

Fill in the blanks to complete each of the following theorem statements:

For all real numbers$x$,role="math" localid="1650909042720" ${\cos }^{2}hx-{\sin }^{2}hx=$----------

Limit calculations: Find each of the following limits.

$\underset{h→0}{\lim }\frac{{\displaystyle \frac{1}{2+h}}-0.5}{h}$

localid="1648442008841" $\underset{z→2}{\lim }\frac{z^2-4}{z-2}$

$\underset{z→4}{\lim }\frac{(1-3z)+11}{z-4}$

$\underset{z→1}{\lim }\frac{{\displaystyle \frac{1}{z}}-1}{z-1}$

For each function k that follows, find functions f , g, and h so that k = f â—¦ g â—¦ h. There may be more than one possible answer.

$$\begin{gathered} k\left(x\right)={\left(x^2+1\right)}^3 \\ k\left(x\right)=\sqrt{1+{\left(x-2\right)}^2} \\ k\left(x\right)=\sqrt{\frac{1}{3x+1}} \\ k\left(x\right)=\frac{1}{\sqrt{3x+1}} \end{gathered}$$

Find each of the following limits.

$\begin{array}{l}\underset{h→0}{\lim }\frac{(1-h)-1}{h}\\ \underset{h→0}{\lim }\frac{\left(3(-1+h{)}^2+1\right)-4}{h}\\ \underset{h→0}{\lim }\frac{\frac{1}{2+h}-0.5}{h}\\ \underset{z→2}{\lim }\frac{z^2-4}{z-2}\\ \underset{z→4}{\lim }\frac{(1-3z)+11}{z-4}\\ \underset{z→1}{\lim }\frac{\frac{1}{z}-1}{z-1}\end{array}$