5.5.2 Derivative of the Tangent Function

In Figure 7 we show again the graph of the tangent function. We can make some observations right away about the derivative of this function:

• It must always be positive.
• Its values must become arbitrarily large near odd multiples of \(\pi/2\).
• It must have a smallest positive value at integer multiples of \(\pi\).
• It must have the same period as the tangent function, namely, \(\pi\).

Figure 7   \(y = \tan \theta\)

Example 2

Calculate \(\frac{d}{d \theta}\tan\theta\).

Solution   We can write \(\tan\theta\) as a product, \((\sin\theta)\times (\cos\theta)^{-1}\), and then apply the Product Rule (and other rules):

d d ⁢ θ ⁢ tan ( θ )	= d d ⁢ θ ⁢ sin ( θ ) [ cos ( θ ) ] - 1
	= ⁢ sin ( θ ) d d ⁢ θ [ cos ( θ ) ] - 1 ⁢ + [ cos ( θ ) ] - 1 d d ⁢ θ ⁢ sin ( θ )
	= ⁢ sin ( θ ) ( - 1 ) [ cos ( θ ) ] - 2 ⁢ [ - sin ( θ ) ] + [ cos ( θ ) ] - 1 cos ( θ )
	= sin 2 ( θ ) cos 2 ( θ ) ⁢ + 1
	= tan 2 ( θ ) ⁢ + 1.

Checkpoint 2

Activity 4

Now that we have a formula for the derivative of the tangent function, confirm from the formula each of the observations at the top of this page:

1. \(\tan^2\theta + 1\) is always positive.
2. The values of \(\tan^2\theta + 1\) are arbitrarily large near odd multiples of \(\pi/2\).
3. \(\tan^2\theta + 1\) has its smallest positive value at integer multiples of \(\pi\). What is that smallest value?
4. \(\tan^2\theta + 1\) has the same period as the tangent function, namely, \(\pi\).
5. Use the information in parts (a)-(d) to sketch a graph of the derivative of the tangent function — freehand, on a sheet of paper.

Comment on Activity 4

The formula for the derivative of the tangent function is usually not written in the form \(\tan^2\theta + 1\). You may — or may not — recall a trigonometric identity that relates this expression to another trigonometric function, the secant. You should recall that the secant is the reciprocal of the cosine:

sec ( θ ) = ⁢ ⁢ 1 cos ( θ ).

Activity 5

Show that

\(\frac{d}{d \theta}\tan\theta = \sec^2\theta\).

In other words, establish the identity

\(\tan^2\theta + 1 = \sec^2\theta\).

Comment on Activity 5