Derivative of tangent evaluated at a point

The derivative of $\tan(x)$ is a standard memorized trigonometric derivative: $h'(x) = \sec^2(x)$. This comes directly from the basic differentiation rules for trig functions, with no algebraic manipulation needed here.

To evaluate at $x = \pi/4$, note that $\sec\left(\frac{\pi}{4}\right) = \frac{1}{\cos(\pi/4)} = \sqrt{2}$. Squaring gives $h'\left(\frac{\pi}{4}\right) = \left(\sqrt{2}\right)^2 = 2$.

The derivative of tangent is secant squared, so the trigonometric function being squared is secant. This directly recalls the memorized rule h prime of x equals secant squared of x used in part a. Sine, cosine, and the other trig functions do not appear in this particular derivative.