Find the derivative of f(x)=`tanx` at x=0.

Answer

To find the derivative of the function \( f(x) = \tan x \) at \( x = 0 \), we can use the definition of the derivative. The derivative of a function \( f \) at a point \( a \) is given by:

\[ f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a} \]

In our case, we want to find \( f'(0) \):

1. **Identify the function and the point**:
- We have \( f(x) = \tan x \) and we want to find \( f'(0) \).

2. **Apply the definition of the derivative**:
\[ f'(0) = \lim_{x \to 0} \frac{f(x) - f(0)}{x - 0} \]

3. **Calculate \( f(0) \)**:
- We know that \( \tan(0) = 0 \), so \( f(0) = 0 \).

4. **Substitute \( f(x) \) and \( f(0) \) into the derivative formula**:
\[ f'(0) = \lim_{x \to 0} \frac{\tan x - 0}{x} = \lim_{x \to 0} \frac{\tan x}{x} \]

5. **Evaluate the limit**:
- We need to find \( \lim_{x \to 0} \frac{\tan x}{x} \). This limit is a standard limit in calculus, and it is known that:
\[ \lim_{x \to 0} \frac{\tan x}{x} = 1 \]

6. **Conclusion**:
- Therefore, we have:
\[ f'(0) = 1 \]

Thus, the derivative of \( f(x) = \tan x \) at \( x = 0 \) is \( 1 \).