Derivatives of other trigonometric functions

Let $h(x) = \sec(x)$ and recall that $\sec(x) = \frac{1}{\cos(x)}$.

1. What is the domain of $h$?

2. Use the quotient rule to develop a formula for $h'(x)$ that is expressed completely in terms of $\sin(x)$ and $\cos(x)$.

3. How can you use other relationships among trigonometric functions to write $h'(x)$ only in terms of $\tan(x)$ and $\sec(x)$?

4. What is the domain of $h'$? How does this compare to the domain of $h$?

Let $p(x) = \csc(x)$ and recall that $\csc(x) = \frac{1}{\sin(x)}$.

1. What is the domain of $p$?

2. Use the quotient rule to develop a formula for $p'(x)$ that is expressed completely in terms of $\sin(x)$ and $\cos(x)$.

3. How can you use other relationships among trigonometric functions to write $p'(x)$ only in terms of $\cot(x)$ and $\csc(x)$?

4. What is the domain of $p'$? How does this compare to the domain of $p$?

Respond to each of the following prompts. Where a derivative is requested, be sure to label the derivative function with its name using proper notation.

1. Let $f(x) = 5 \sec(x) - 2\csc(x)$. Find the slope of the tangent line to $f$ at the point where $x =\frac{\pi}{3}$.

2. Let $p(z) = z^2\sec(z) - z\cot(z)$. Find the instantaneous rate of change of $p$ at the point where $z = \frac{\pi}{4}$.

3. Let $h(t) = \displaystyle \frac{\tan (t)}{t^2+1} - 2e^t \cos(t)$. Find $h'(t)$.

4. Let $g(r) = \displaystyle \frac{r \sec(r) }{5^r}$. Find $g'(r)$.

5. When a mass hangs from a spring and is set in motion, the object's position oscil…

An object moving vertically has its height at time $t$ (measured in feet, with time in seconds) given by the function $h(t) = 3 + \frac{2\cos(t)}{1.2^t}$.

1. What is the object's instantaneous velocity when $t =2$?

2. What is the object's acceleration at the instant $t = 2$?

3. Describe in everyday language the behavior of the object at the instant $t = 2$.

Let $f(x) = \sin(x) \cot(x)$.

1. Use the product rule to find $f'(x)$.

2. True or false: for all real numbers $x$, $f(x) = \cos(x)$.

3. Explain why the function that you found in (a) is almost the opposite of the sine function, but not quite. (Hint: convert all of the trigonometric functions in (a) to sines and cosines, and work to simplify. Think carefully about the domain of $f$ and the domain of $f'$.)

Let $p(z)$ be given by the rule .

1. Determine $p'(z)$.

2. Find an equation for the tangent line to $p$ at the point where $z = 0$.

3. At $z = 0$, is $p$ increasing, decreasing, or neither? Why?