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The question is to find the derivative of the function \( y = 12x^3 - 10x^5 \), and then select the correct answer from the options provided.

First, let's find the derivative \( \frac{dy}{dx} \):

\[ y = 12x^3 - 10x^5 \]

The derivative of \( y \) with respect to \( x \) is:

\[ \frac{dy}{dx} = \frac{d}{dx}(12x^3) - \frac{d}{dx}(10x^5) \]

Using the power rule \( \frac{d}{dx}(x^n) = nx^{n-1} \):

\[ \frac{d}{dx}(12x^3) = 12 \cdot 3x^{3-1} = 36x^2 \]
\[ \frac{d}{dx}(10x^5) = 10 \cdot 5x^{5-1} = 50x^4 \]

So, the derivative is:

\[ \frac{dy}{dx} = 36x^2 - 50x^4 \]

Now, let's compare this with the given options:

1. \( 36x^4 - 10x^4 \)
2. \( 72x^2 - 20x^3 \)
3. \( 10x^4 - 36x^2 \)
4. \( 12x^2 - 10x^5 \)

The correct answer is none of the options directly. However, it seems there might be a typographical error in the provided options. The closest option to our derived result is \( 36x^2 - 10x^4 \), but it should be \( 36x^2 - 50x^4 \).

Would you like further details or have any questions?

Here are some related questions to deepen your understanding:

1. What is the power rule for differentiation?
2. How do you differentiate polynomial functions?
3. Can you explain the difference between a polynomial and a monomial?
4. What are the common mistakes to avoid while differentiating?
5. How can you verify the correctness of a derivative?
6. What is the significance of higher-order derivatives?
7. How do derivatives apply to real-world problems?
8. Can you find the second derivative of the given function \( y = 12x^3 - 10x^5 \)?

**Tip:** Always double-check your differentiation steps and the resulting expressions for accuracy, especially when dealing with higher-order terms.

Learn how to find the derivative of y = 12x^3 - 10x^5 with a step-by-step solution. This problem involves differentiating polynomial functions using the power rule. Suitable for advanced high school students in Grades 11-12.

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