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To find the derivative of the function \( y = e^{6x^5} \), we will use the chain rule.

1. The outer function is \( e^u \), where \( u = 6x^5 \). The derivative of \( e^u \) with respect to \( u \) is just \( e^u \).
   
2. The inner function is \( u = 6x^5 \), and the derivative of \( 6x^5 \) with respect to \( x \) is \( 30x^4 \).

Now, applying the chain rule:

\[
\frac{dy}{dx} = e^{6x^5} \cdot \frac{d}{dx}(6x^5)
\]

\[
\frac{dy}{dx} = e^{6x^5} \cdot 30x^4
\]

So the derivative is:

\[
y' = 30x^4 e^{6x^5}
\]

Would you like any further explanation on the steps? Here are 5 related questions for deeper understanding:

1. How would you apply the chain rule to other exponential functions?
2. What is the derivative of \( y = e^{2x^3} \)?
3. How do you handle more complicated inner functions when using the chain rule?
4. What happens when differentiating natural logarithmic functions instead of exponential ones?
5. Can you explain the difference between the product rule and the chain rule?

**Tip:** When applying the chain rule, always identify the "outer" and "inner" functions clearly to avoid confusion.

Find the derivative of the function: y = e^(6x^5)

Find the derivative of the function y = e^(6x^5) using the chain rule. This problem covers exponential differentiation and applying the chain rule. Suitable for advanced calculus students in Grades 11-12.

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