try to guess a formula for f'(x) where f(x)  = 2^x

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.

try to guess a formula for f'(x) where f(x) = 2^x

Discover how to differentiate the exponential function f(x) = 2^x. Using logarithmic differentiation, learn to apply the formula f'(x) = 2^x * ln(2). This explanation covers key concepts in calculus, suitable for students in Grades 11-12.

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