91Ó°ÊÓ

Calculate each of the derivatives or derivative values in Exercises 47–52.

$\frac{d^2}{d{x}^2}\left({\left(x\sqrt{x+1}\right)}^{-2}\right)$

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.

$f\left(x\right)=\frac{x^4-7x^3}{2x}$

Calculate each of the derivatives or derivative values in Exercises 47–52.

$\frac{d^2}{d{x}^2}\left(\frac{{\displaystyle \frac{1}{x}-3x^2}}{x^5-{\displaystyle \frac{1}{\sqrt{x}}}}\right)$

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

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

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

$f\left(x\right)=x^{-2}{e}^{4x}{\sin }^{-1}x$

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:

When the bowling ball hits the ground, with$h=-0.5,h=-0.2andh=-0.1$

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

$f\left(x\right)=\left\{\frac{1}{2}-\frac{1}{4}x,x≤0;\frac{1}{1+e^x},x>0\right\}$

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

$f\left(x\right)=\sec \left(1+{\tan }^{-1}x\right)$

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.

$f\left(x\right)={\mathrm{Ï€}}^2$

Think about what you did today and how far north you were from your house or dorm throughout the day. Sketch a graph that represents your distance north from your house or dorm over the course of the day, and explain how the graph reflects what you did today. Then sketch a graph of your velocity.