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.