91Ó°ÊÓ

Find the derivatives of the functions in Exercises 21–46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

$f\left(x\right)=\sqrt{1+\sqrt{1+\sqrt{1-2x}}}$

A bowling ball dropped from a height of $400$feet will be $s\left(t\right)=400-16t^2$feet from the ground after $t$seconds. Use a sequence of average velocities to estimate the instantaneous velocities described below:

After $t=1$ seconds, with $h=0.5,h=0.25,h=-0.5andh=-0.2$

Describe the derivatives of each of the piecewise-defined functions in Exercises 45–48.

$f\left(x\right)=\left\{\ln \right(-x),\mathrm{if}x<0;\ln x,\mathrm{if}xâ‰\yen 0\}$

Use the differentiation rules developed in this section to find the derivatives of the functions in Exercises 35-64. Note that it may be necessary to do some preliminary algebra before differentiating.

$f\left(x\right)=\frac{x}{x+1}$

Use the definition of the derivative to find $f\text{'}$for each function $f$in Exercises 34-59

$f\left(x\right)=\frac{1}{x^3}$

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

$f\left(x\right)=\ln \left(arcsec\left({\sin }^{2}x\right)\right)$

Use the definition of the derivative to find $f\text{'}$for each function $f$in Exercises 34-59

$f\left(x\right)=3\sqrt{x}$

A bowling ball dropped from a height of $400$feet will be $s\left(t\right)=400-16t^2$feet from the ground after $t$seconds. Use a sequence of average velocities to estimate the instantaneous velocities described below:

After $t=2$ seconds, with $h=0.1,h=0.01h=-0.1andh=-0.01$

Describe the derivatives of each of the piecewise-defined functions in Exercises 45–48.

$f\left(x\right)=\{x^2,x<1;1+\ln x,xâ‰\yen 1\}$

Use the differentiation rules developed in this section to find the derivatives of the functions in Exercises 35-64. Note that it may be necessary to do some preliminary algebra before differentiating.

$f\left(x\right)=\frac{x^2-1}{x+1}$