State the unit associated with the following derivatives and state the physical meaning of each derivative. $w^{\prime}(t)$

Determine the derivative of the function given below.

$y=(\sec x+\tan x)^5$

Determine the derivative of the function given below.

$y=\cos ^4 \theta+\sin ^4 \theta$

Calculate the following equation which is given by $\frac { d } { d x } \left( x ^ { 2 } + x \right) ^ { 2 }$using the Chain Rule. Simplify your answer.

A small probe is launched vertically from the ground. After it reaches its high point, a parachute deploys and the probe descends to Earth. The height of the probe above ground is $s(t)=\frac{300 t-50 t^{2}}{t^{3}+2}$, for $0 \leq t \leq 6$. a. Graph the height function and describe the motion of the probe. b. Find the velocity of the probe. c. Graph the velocity function and determine the approximate time at which the velocity is a maximum.

The following table shows the average body mass m(t) (in grams) and average wing chord length w(t) (in millimeters), along with the derivatives m'(t) and w'(t), oft-week-old Indian spotted owlets. The flight formula function f(t)=w(t)/m(t), which is the ratio of wing chord length to mass, is used to predict when these fledglings are old enough to fly. The values off are less than 1, but approach 1 as t increases. When f is close to 1, the fledglings are capable of flying, which is important for determining when rescued fledglings can be released back into the wild.

$$\begin{array}{ccccc}{t} & {m(t)} & {m^{\prime}(t)} & {w(t)} & {w^{\prime}(t)} \\ \hline 1 & {23.32} & {41.45} & {10.14} & {14.5} \\ {1.5} & {50.59} & {64.94} & {20.13} & {26.17} \\ {2} & {82.83} & {57.95} & {36.7} & {39.86} \\ {2.5} & {105.13} & {31.08} & {58.92} & {47.38} \\ {3} & {119.76} & {12.48} & {98.99} & {27.94} \\ {4} & {121.37} & {0.51} & {118.34} & {15.74} \\ {4.5} & {121.37} & {0.17} & {118.34} & {3.85} \\ {5} & {121.42} & {0.06} & {119.34} & {1.8} \\ {6} & {121.44} & {0.02} & {120.32} & {0.84} \\ {6.5} & {121.45} & {0.01} & {120.61} & {0.38}\end{array}$$

State the units associated with the following derivatives and state the physical meaning of each derivative. a. m'(t). b. w'(t). c. f'(t).

The following limit $\lim _ { x \rightarrow 1 } \frac { x ^ { 100 } - 1 } { x - 1 }$ represent f'(a) for some function f and some real number a. Find a possible function f and number a.

Normal lines A normal line on a curve passes through a point $P$ on the curve perpendicular to the line tangent to the curve at $P$. Use the following equations and graphs to determine an equation of the normal line at the given point and illustrate your work by graphing the curve with the normal line.
previous Exercise

Is f(x)=

$$\frac { x ^ { 2 } - 5 x + 6 } { x - 2 }$$

differentiable at x=2? Justify your answer.

For the function

$$f(x)= \begin{cases}x & \text { if } x \leq 0 \\ x+1 & \text { if } x>0 .\end{cases}$$

Is $f$ differentiable at 0 ? Explain.

For the function

$$f(x)= \begin{cases}x & \text { if } x \leq 0 \\ x+1 & \text { if } x>0 .\end{cases}$$

Graph $f^{\prime}$ on its domain.

For the function

$$f(x)= \begin{cases}x & \text { if } x \leq 0 \\ x+1 & \text { if } x>0 .\end{cases}$$

For $x>0$, what is $f^{\prime}(x)$ ?

For the function

$$f(x)= \begin{cases}x & \text { if } x \leq 0 \\ x+1 & \text { if } x>0 .\end{cases}$$

For $x<0$, what is $f^{\prime}(x)$ ?

a. Graph the function

$$f(x)=\left\{\begin{array}{ll}{x} & {\text { if } x \leq 0} \\ {x+1} & {\text { if } x>0}\end{array}\right.$$

. b. For x<0, what is f'(x)? c. For x>0, what is f'(x)? d. Graph f' on its domain. e. Is f differentiable at 0? Explain.