2.4: Derivatives of Other Trigonometric Functions

Derivatives of the cotangent, secant, and cosecant functions

In Preview Activity $2.4.1$ , we found that the derivative of the tangent function can be expressed in several ways, but most simply in terms of the secant function. Next, we develop the derivative of the cotangent function.

Let $g(x) = \cot(x)\text{.}$ To find $g'(x)\text{,}$ we observe that $g(x) = \frac{\cos(x)}{\sin(x)}$ and apply the quotient rule. Hence

By the Fundamental Trigonometric Identity, we see that $g'(x) = -\frac{1}{\sin^2(x)}\text{,}$ and recalling that $\csc(x) = \frac{1}{\sin(x)}\text{,}$ it follows that we can express $g'$ by the rule

Note that neither $g$ nor $g'$ is defined when $\sin(x) = 0\text{,}$ which occurs at every integer multiple of $\pi\text{.}$ Hence we have the following rule.

Cotangent Function

For all real numbers $x$ such that $x \ne k\pi\text{,}$ where $k = 0, \pm 1, \pm 2, \ldots\text{,}$

$$\dfrac{d}{dx} [\cot(x)] = -\csc^2(x)\text{.}$$

Notice that the derivative of the cotangent function is very similar to the derivative of the tangent function we discovered in Preview Activity $2.4.1$.

Tangent Function

For all real numbers $x$ such that $x \ne \frac{(2k+1)\pi}{2}\text{,}$ where $k = \pm 1, \pm 2, \ldots\text{,}$

$$\dfrac{d}{dx} [\tan(x)] = \sec^2(x)\text{.}$$

In the next two activities, we develop the rules for differentiating the secant and cosecant functions.

Activity $2.4.2$

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

1. What is the domain of $h\text{?}$
2. Use the quotient rule to develop a formula for $h'(x)$ that is expressed completely in terms of $\sin(x)$ and $\cos(x)\text{.}$
3. How can you use other relationships among trigonometric functions to write $h'(x)$ only in terms of $\tan(x)$ and $\sec(x)\text{?}$
4. What is the domain of $h'\text{?}$ How does this compare to the domain of $h\text{?}$

Activity $2.4.3$

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

1. What is the domain of $p\text{?}$
2. Use the quotient rule to develop a formula for $p'(x)$ that is expressed completely in terms of $\sin(x)$ and $\cos(x)\text{.}$
3. How can you use other relationships among trigonometric functions to write $p'(x)$ only in terms of $\cot(x)$ and $\csc(x)\text{?}$
4. What is the domain of $p'\text{?}$ How does this compare to the domain of $p\text{?}$

Using the quotient rule we have determined the derivatives of the tangent, cotangent, secant, and cosecant functions, expanding our overall library of functions we can differentiate. Observe that just as the derivative of any polynomial function is a polynomial, and the derivative of any exponential function is another exponential function, so it is that the derivative of any basic trigonometric function is another function that consists of basic trigonometric functions. This makes sense because all trigonometric functions are periodic, and hence their derivatives will be periodic, too.

The derivative retains all of its fundamental meaning as an instantaneous rate of change and as the slope of the tangent line to the function under consideration.

Activity $2.4.4$

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

1. Let $f(x) = 5 \sec(x) - 2\csc(x)\text{.}$ Find the slope of the tangent line to $f$ at the point where $x =\frac{\pi}{3}\text{.}$
2. Let $p(z) = z^2\sec(z) - z\cot(z)\text{.}$ Find the instantaneous rate of change of $p$ at the point where $z = \frac{\pi}{4}\text{.}$
3. Let $h(t) = \displaystyle \frac{\tan (t)}{t^2+1} - 2e^t \cos(t)\text{.}$ Find $h'(t)\text{.}$
4. Let $g(r) = \displaystyle \frac{r \sec(r) }{5^r}\text{.}$ Find $g'(r)\text{.}$
5. When a mass hangs from a spring and is set in motion, the object's position oscillates in a way that the size of the oscillations decrease. This is usually called a damped oscillation. Suppose that for a particular object, its displacement from equilibrium (where the object sits at rest) is modeled by the function
   $$s(t) = \dfrac{15 \sin(t)}{e^t}\text{.}$$

   Assume that $s$ is measured in inches and $t$ in seconds. Sketch a graph of this function for $t \ge 0$ to see how it represents the situation described. Then compute $ds/dt\text{,}$ state the units on this function, and explain what it tells you about the object's motion. Finally, compute and interpret $s'(2)\text{.}$

Summary

• The derivatives of the other four trigonometric functions are
  $$\frac{d}{dx}[\tan(x)] = \sec^2(x), \ \ \frac{d}{dx}[\cot(x)] = -\csc^2(x)\text{,}$$
  $$\frac{d}{dx}[\sec(x)] = \sec(x)\tan(x), \ \text{and} \ \frac{d}{dx}[\csc(x)] = -\csc(x)\cot(x)\text{.}$$

  Each derivative exists and is defined on the same domain as the original function. For example, both the tangent function and its derivative are defined for all real numbers $x$ such that $x \ne \frac{k\pi}{2}\text{,}$ where $k = \pm 1, \pm 2, \ldots\text{.}$

• The four rules for the derivatives of the tangent, cotangent, secant, and cosecant can be used along with the rules for power functions, exponential functions, and the sine and cosine, as well as the sum, constant multiple, product, and quotient rules, to quickly differentiate a wide range of different functions.