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?

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.