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Let's differentiate the function \( y = 8e^x + \frac{7}{\sqrt[3]{x}} \).

### Step 1: Differentiate \( 8e^x \)
The derivative of \( e^x \) is \( e^x \). Thus:
\[
\frac{d}{dx}(8e^x) = 8e^x
\]

### Step 2: Differentiate \( \frac{7}{\sqrt[3]{x}} \)
We can rewrite \( \frac{7}{\sqrt[3]{x}} \) as \( 7x^{-\frac{1}{3}} \). Using the power rule \( \frac{d}{dx}(x^n) = nx^{n-1} \), we get:
\[
\frac{d}{dx}(7x^{-\frac{1}{3}}) = 7 \times \left(-\frac{1}{3}\right) x^{-\frac{1}{3} - 1} = -\frac{7}{3} x^{-\frac{4}{3}}
\]

### Final Result:
Now combining the two derivatives, we get:
\[
y' = 8e^x - \frac{7}{3}x^{-\frac{4}{3}}
\]

So, the derivative is:
\[
y' = 8e^x - \frac{7}{3\sqrt[3]{x^4}}
\]

Let me know if you want further explanation or details!

### Related questions:
1. What is the derivative of \( e^{2x} \)?
2. How do you apply the chain rule to a function like \( e^{f(x)} \)?
3. What is the power rule for derivatives and how is it used?
4. Can you explain how to differentiate \( x^{1/n} \)?
5. How do logarithmic functions influence differentiation?

### Tip:
Always rewrite complicated terms like roots or reciprocals as powers of \( x \) before differentiating for easier application of the rules.

Differentiate the function y = 8e^x + 7/∛x

Learn how to differentiate the function y = 8e^x + 7/∛x using key differentiation rules such as the power rule and the derivative of exponential functions. This step-by-step solution covers advanced calculus concepts, ideal for students in Grades 11-12.

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