To differentiate the function \( e^{\frac{2x}{3}} \) with respect to \( x \), we can apply the chain rule.

Given:
\[
f(x) = e^{\frac{2x}{3}}
\]

The derivative of \( f(x) \) with respect to \( x \) is:

\[
f'(x) = \frac{d}{dx} \left(e^{\frac{2x}{3}}\right)
\]

Using the chain rule, we first differentiate the exponent with respect to \( x \):

\[
\frac{d}{dx}\left(\frac{2x}{3}\right) = \frac{2}{3}
\]

Now, differentiate the exponential function:

\[
\frac{d}{dx}\left(e^{\frac{2x}{3}}\right) = e^{\frac{2x}{3}} \cdot \frac{2}{3}
\]

Thus, the derivative is:

\[
f'(x) = \frac{2}{3} \cdot e^{\frac{2x}{3}}
\]

Would you like more details or have any questions?

Here are some related questions you might find useful:
1. What is the derivative of \( e^{kx} \) where \( k \) is a constant?
2. How do you apply the chain rule to more complex functions?
3. What is the integral of \( e^{\frac{2x}{3}} \)?
4. How do you differentiate functions with multiple variables?
5. What is the significance of the exponential function in calculus?

**Tip:** Remember, the chain rule is essential for differentiating composite functions where one function is nested within another.

Learn how to differentiate the function e^(2x/3) using the chain rule. This detailed guide provides a step-by-step solution, including formulas and key concepts of exponential functions and the chain rule. Suitable for college-level mathematics students.

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