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using the images for info, can you guess the formula for f'(x) = 2^x

The image shows the graph of the exponential function \( f(x) = 2^x \), and you are asking for the derivative, \( f'(x) \).

To differentiate \( f(x) = 2^x \), we use the chain rule and the fact that the derivative of \( a^x \) (for any base \( a \)) is given by:
\[
\frac{d}{dx}(a^x) = a^x \ln(a)
\]
where \( \ln(a) \) is the natural logarithm of the base \( a \).

In the case of \( f(x) = 2^x \), the derivative is:
\[
f'(x) = 2^x \ln(2)
\]

Thus, the derivative of \( f(x) = 2^x \) is \( f'(x) = 2^x \ln(2) \).

Would you like further explanation or have questions about this? 

Here are 5 related questions to expand on the concept:
1. How do you differentiate more complex exponential functions like \( 3^x \)?
2. What is the difference between differentiating \( 2^x \) and \( e^x \)?
3. How does the graph of the derivative \( f'(x) = 2^x \ln(2) \) compare to the graph of \( f(x) = 2^x \)?
4. Can you explain the chain rule in detail when applied to functions like \( 2^x \)?
5. How do logarithmic functions relate to the differentiation of exponential functions?

**Tip:** When differentiating exponential functions with bases other than \( e \), always remember to multiply by the natural logarithm of the base!

Using the image, can you guess the formula for f'(x) = 2^x?

Explore how to differentiate the exponential function f(x) = 2^x to find f'(x). This problem uses the chain rule and the formula for the derivative of a^x. Suitable for students in Grades 11-12 learning advanced calculus concepts.

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