• We will examine the derivatives of the tangent function and the reciprocal trigonometric functions graphically.
• We will use the known derivatives of the sine and cosine functions to verify formulas for each of the derivatives.

Consider the graph of the tangent function.

What does the graph of \(f'(x)\) look like? What information can we gather from the graph of \(f(x)=\tan(x)\)?

• Since \(f(x) = \tan(x)\) is undefined at the points \(x = \pm \dfrac{\pi}{2}, \pm \dfrac{3\pi}{2}, \ldots,\) the derivative \(f'(x)\) will be undefined at each of these points as well.
• The slope of \(f(x)\) is always positive and so we have \(f'(x) \gt 0\) for the entire domain of \(f(x)\).
• The slope appears flattest (closest to horizontal) at the points \(x=0, \pm \pi, \pm 2\pi \ldots\) and so we might expect \(f'(x)\) to attain some minimal value at these points.
• The slope is approaching \( + \infty\) as \(x\) approaches any of the vertical asymptotes at \(x = \pm \dfrac{\pi}{2}, \pm \dfrac{3\pi}{2}, \ldots\)

We can use these observations to draw a rough sketch of the derivative \(f'(x)\). Try this now. Does this sketched curve remind you of any known function?

Hopefully, the graph that you sketched looked something like the following:

Does the shape of this curve appear similar to the shape of another trigonometric function?

Consider the graph of the function \(g(x) = \sec(x)\).

The graph of \(g(x)\) and \(f'(x)\) share the same vertical asymptotes, but there are two observable differences in the graphs. The most obvious one is that the function \(g(x)\) takes on negative values, while \(f'(x)\) is always positive. What does this suggest? Absolute value of \(g(x)\)? The less obvious difference is that the the function \(f'(x)\) increases faster than the function \(g(x)\). This suggests that \(f'(x)\) is not the absolute value of \(g(x)\), but perhaps it is the square of \(g(x)\). From the graphical evidence, we suspect that \(f'(x) = (g(x))^{2} = \sec^{2}(x)\).

Since \(\tan(x) = \dfrac{\sin(x)}{\cos(x)}\) and the derivatives of \(\sin(x)\) and \(\cos(x)\) are known, we can confirm this suspicion using the quotient rule.

Find the derivatives of the functions \(f(x) = \tan(3x^{2}+4)\) and \(g(x) = \tan(x)\sin(x)\).

Solution

Since \(\dfrac{d}{du}\Big(\tan(u)\Big) = \sec^{2}(u)\) then, using the chain rule, we get

By the product rule, we have