https://file.aiquickdraw.com/mathbot/file/1729903878937v4e1uejj.jpg

The function provided in the image is:

\[
y = 2e^x + \frac{7}{\sqrt[3]{x}}
\]

We need to differentiate this function with respect to \( x \).

### Step-by-Step Solution

1. **Differentiate \( 2e^x \):**
   The derivative of \( 2e^x \) with respect to \( x \) is:
   \[
   \frac{d}{dx}(2e^x) = 2e^x
   \]

2. **Differentiate \( \frac{7}{\sqrt[3]{x}} \):**
   Rewrite \( \frac{7}{\sqrt[3]{x}} \) as \( 7x^{-\frac{1}{3}} \).

   Then, use the power rule to find the derivative of \( 7x^{-\frac{1}{3}} \):
   \[
   \frac{d}{dx}\left(7x^{-\frac{1}{3}}\right) = 7 \cdot -\frac{1}{3}x^{-\frac{4}{3}} = -\frac{7}{3}x^{-\frac{4}{3}}
   \]

3. **Combine the results:**
   Now, we can combine the derivatives:
   \[
   y' = 2e^x - \frac{7}{3}x^{-\frac{4}{3}}
   \]

### Final Answer

\[
y' = 2e^x - \frac{7}{3x^{\frac{4}{3}}}
\]

Would you like more details on any of the steps, or do you have any other questions?

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Here are five related questions to explore:

1. How would you differentiate a function with a variable in both the numerator and denominator?
2. What are the steps to differentiate composite functions like \( e^{x^2} \)?
3. How can you use implicit differentiation on functions not explicitly solved for \( y \)?
4. What techniques would be useful for differentiating logarithmic functions?
5. How does the power rule apply when the exponent is negative or a fraction?

**Tip:** Remember, for fractional exponents, rewriting the function in terms of powers makes differentiation more straightforward.

Differentiate the function y = 2e^x + 7/x^(1/3)

Find the derivative of the function y = 2e^x + 7/x^(1/3) using the power rule and properties of exponential functions. This calculus problem is suitable for students in Grades 11-12, covering essential differentiation techniques.

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