Let’s start with the derivative of the function \( f(x) = \tan(x) \). The limit of the difference quotient is:
\begin{align} f'(x) &= \lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0} \\ &=\lim_{x \to x_0} \frac{\tan(x) - \tan(x_0)}{x - x_0} \end{align}
Using the identity for the difference of tangents:
\[ \tan(x) - \tan(x_0) = \frac{\sin(x - x_0)}{\cos(x) \cos(x_0)} \]
Substituting this identity into the difference quotient, we get:
\[ \frac{\tan(x) - \tan(x_0)}{x - x_0} = \frac{\frac{\sin(x - x_0)}{\cos(x) \cos(x_0)}}{x - x_0} \]
Simplifying:
\[ \frac{1}{\cos(x) \cos(x_0)} \cdot \frac{\sin(x - x_0)}{x - x_0} \]
Now, as \( x \to x_0 \), we use the notable limit:
\[ \lim_{x \to x_0} \frac{\sin(x - x_0)}{x - x_0} = 1 \]
Therefore, the expression becomes:
\[ \lim_{x \to x_0} \frac{1}{\cos(x) \cos(x_0)} \cdot 1 = \frac{1}{\cos^2(x_0)} \]
Since \(\sec(x) = \frac{1}{\cos(x)}\), we can rewrite the final result as:
\[ \lim_{x \to x_0} \frac{\tan(x) - \tan(x_0)}{x - x_0} = \sec^2(x_0) \]
Therefore:
\[ f'(x) = \sec^2(x) \quad , \quad \forall x \in \mathbb{R} \setminus \left\{ \frac{\pi}{2} + k\pi \mid k \in \mathbb{Z} \right\} \]
Now, let’s compute the derivative of the function \( g(x) = \cot(x) \). The limit of the difference quotient is:
\begin{align} g'(x) &= \lim_{x \to x_0} \frac{g(x) - g(x_0)}{x - x_0} \\ &=\lim_{x \to x_0} \frac{\cot(x) - \cot(x_0)}{x - x_0} \end{align}
Using the identity for the difference of cotangents:
\[ \cot(x) - \cot(x_0) = -\frac{\sin(x - x_0)}{\sin(x) \sin(x_0)} \]
Substituting this identity into the difference quotient, we get:
\[ \frac{\cot(x) - \cot(x_0)}{x - x_0} = \frac{-\frac{\sin(x - x_0)}{\sin(x) \sin(x_0)}}{x - x_0} \]
Simplifying:
\[ -\frac{1}{\sin(x) \sin(x_0)} \cdot \frac{\sin(x - x_0)}{x - x_0} \]
Now, as \( x \to x_0 \), we use the notable limit:
\[ \lim_{x \to x_0} \frac{\sin(x - x_0)}{x - x_0} = 1 \]
Therefore, the expression becomes:
\[ \lim_{x \to x_0} -\frac{1}{\sin(x) \sin(x_0)} \cdot 1 = -\frac{1}{\sin^2(x_0)} \]
Since \(\csc(x) = \frac{1}{\sin(x)}\), we can rewrite the final result as:
\[ \lim_{x \to x_0} \frac{\cot(x) - \cot(x_0)}{x - x_0} = -\csc^2(x_0) \]
Therefore:
\[ g'(x) = -\csc^2(x)\quad , \quad \forall x \in \mathbb{R} \setminus \{ k\pi \mid k \in \mathbb{Z} \} \]