In Exercises , use the definition of the derivative to find $f^{\prime}(x)$.

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

In Exercises, a point is moving along the graph of the function such that $d x / d t$ is 2 centimeters per second. Find $d y / d t$ for the specified values of $x$.

$\underline{\text{ Function $\quad$ Values of }}$ $x$

$y=\sin x$ , $x=\frac{\pi}{3}$

In Exercises, a point is moving along the graph of the function such that $d x / d t$ is 2 centimeters per second. Find $d y / d t$ for the specified values of $x$.

$\underline{\text{ Function $\quad$ Values of }}$ $x$

$y=\sin x$ , $x=\frac{\pi}{4}$

In Exercises, a point is moving along the graph of the function such that $d x / d t$ is 2 centimeters per second. Find $d y / d t$ for the specified values of $x$.

$\underline{\text{ Function $\quad$ Values of }}$ $x$

$y=\sin x$ , $x=\frac{\pi}{6}$

Find the tangent line(s) to the curve $y=x^3-9 x$ through the point $(1,-9)$.

Stopping Distance A car is traveling at a rate of 66 feet per second (45 miles per hour) when the brakes are applied. The position function for the car is

$$s(t)=-8.25 t^2+66 t$$

where $s$ is measured in feet and $t$ is measured in seconds. Use this function to complete the table, and find the average velocity during each time interval.

In Exercises , use the result of Exercise 86 to find the derivative of the function.

$h(x)=|x| \cos x$

Acceleration An automobile's velocity starting from rest is $v(t)=\frac{100 t}{2 t+15}$ where $v$ is measured in feet per second. Find the acceleration at each of the following times.

20 seconds

Acceleration An automobile's velocity starting from rest is $v(t)=\frac{100 t}{2 t+15}$ where $v$ is measured in feet per second. Find the acceleration at each of the following times.

10 seconds

Acceleration An automobile's velocity starting from rest is $v(t)=\frac{100 t}{2 t+15}$ where $v$ is measured in feet per second. Find the acceleration at each of the following times.

5 seconds

Let $(a, b)$ be an arbitrary point on the graph of $y=1 / x, x>0$. Prove that the area of the triangle formed by the tangent line through $(a, b)$ and the coordinate axes is 2 .

In Exercises , use the result of Exercise 86 to find the derivative of the function.

$f(x)=\left|x^2-4\right|$

In Exercises , use the result of Exercise 86 to find the derivative of the function.

$$g(x)=|2 x-3|$$

Acceleration The velocity of an object in meters per second is

$$v(t)=36-t^2, \quad 0 \leq t \leq 6 .$$

Find the velocity and acceleration of the object when $t=3$. What can be said about the speed of the object when the velocity and acceleration have opposite signs?