To find the derivative of $f(x) = 2^x$, we can follow these steps using logarithmic differentiation:

Step-by-Step Derivation:

1. Start with $f(x) = 2^x$.

2. Recall that the derivative of an exponential function $a^x$ is given by the formula: $\frac{d}{dx}(a^x) = a^x \cdot \ln(a)$ where $\ln(a)$ is the natural logarithm of the base $a$.

3. Applying this to $f(x) = 2^x$, we have: $f'(x) = 2^x \cdot \ln(2)$

Final Answer:

So, the derivative of $f(x) = 2^x$ is: $f'(x) = 2^x \cdot \ln(2)$

Would you like more details on this process or have any other questions?

Here are 5 related questions to expand your understanding:

1. How is the derivative of other exponential functions like $e^x$ derived?
2. What is the derivative of $f(x) = a^x$ where $a$ is a constant?
3. How does the chain rule apply to the derivative of composite exponential functions?
4. Can you find the second derivative of $f(x) = 2^x$?
5. How does the derivative of logarithmic functions relate to this?

Tip: When differentiating exponential functions with any base $a$, always multiply by $\ln(a)$ to account for the base's growth rate.