To derive the formulas for the derivatives of the given functions $\sec x$, $\csc x$, and $\cot x$, we will use the basic differentiation rules and trigonometric identities.

(a) Derivative of $\sec x$

Recall that $\sec x = \frac{1}{\cos x}$.

Using the quotient rule: $\frac{d}{dx} \sec x = \frac{d}{dx} \left( \frac{1}{\cos x} \right)$ Let $u = 1$ and $v = \cos x$, then: $\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{u'v - uv'}{v^2}$ Here, $u' = 0$ and $v' = -\sin x$. So: $\frac{d}{dx} \sec x = \frac{0 \cdot \cos x - 1 \cdot (-\sin x)}{(\cos x)^2} = \frac{\sin x}{\cos^2 x} = \sec x \tan x$

(b) Derivative of $\csc x$

Recall that $\csc x = \frac{1}{\sin x}$.

Using the quotient rule: $\frac{d}{dx} \csc x = \frac{d}{dx} \left( \frac{1}{\sin x} \right)$ Let $u = 1$ and $v = \sin x$, then: $\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{u'v - uv'}{v^2}$ Here, $u' = 0$ and $v' = \cos x$. So: $\frac{d}{dx} \csc x = \frac{0 \cdot \sin x - 1 \cdot \cos x}{(\sin x)^2} = \frac{-\cos x}{\sin^2 x} = -\csc x \cot x$

(c) Derivative of $\cot x$

Recall that $\cot x = \frac{\cos x}{\sin x}$.

Using the quotient rule: $\frac{d}{dx} \cot x = \frac{d}{dx} \left( \frac{\cos x}{\sin x} \right)$ Let $u = \cos x$ and $v = \sin x$, then: $\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{u'v - uv'}{v^2}$ Here, $u' = -\sin x$ and $v' = \cos x$. So: $\frac{d}{dx} \cot x = \frac{(-\sin x) \sin x - \cos x \cos x}{(\sin x)^2} = \frac{-\sin^2 x - \cos^2 x}{\sin^2 x} = \frac{-(\sin^2 x + \cos^2 x)}{\sin^2 x} = \frac{-1}{\sin^2 x} = -\csc^2 x$

In summary:

1. $\frac{d}{dx} \sec x = \sec x \tan x$
2. $\frac{d}{dx} \csc x = -\csc x \cot x$
3. $\frac{d}{dx} \cot x = -\csc^2 x$

Would you like any details or have any questions?

Here are five related questions to expand on this topic:

1. What is the derivative of $\tan x$?
2. How can you derive the second derivative of $\sec x$?
3. What is the integral of $\sec x \tan x$?
4. Can you find the derivative of $\sec^2 x$?
5. How would you apply the chain rule to find the derivative of $\sec(ax + b)$?

Tip: Always remember to simplify trigonometric expressions using identities when working with derivatives and integrals to make calculations easier.