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To solve the problem of finding \( f'(1) \) for the given function \( f(x) = 5x^4 - 12x^2 - 3x + 1 \), we need to follow these steps:

### Step 1: Find the derivative of \( f(x) \)
The function is:
\[
f(x) = 5x^4 - 12x^2 - 3x + 1
\]
We differentiate each term with respect to \( x \):
- The derivative of \( 5x^4 \) is \( 20x^3 \).
- The derivative of \( -12x^2 \) is \( -24x \).
- The derivative of \( -3x \) is \( -3 \).
- The derivative of the constant \( 1 \) is \( 0 \).

Thus, the derivative \( f'(x) \) is:
\[
f'(x) = 20x^3 - 24x - 3
\]

### Step 2: Evaluate the derivative at \( x = 1 \)
Substitute \( x = 1 \) into the derivative:
\[
f'(1) = 20(1)^3 - 24(1) - 3
\]
Simplifying:
\[
f'(1) = 20 - 24 - 3 = -7
\]

### Final Answer
The correct answer is \( -7 \).

Would you like more details or have any questions? 

Here are some related questions to consider:
1. How do you find the derivative of a polynomial function?
2. What are the rules for differentiating functions?
3. What is the significance of evaluating a derivative at a specific point?
4. How would you find the second derivative of the given function?
5. What does the derivative tell you about the behavior of the function at \( x = 1 \)?
6. How would the derivative change if the function had an additional term, like \( +4x^3 \)?
7. What is the meaning of the first derivative in terms of the slope of the tangent line?
8. How does the power rule for differentiation work?

**Tip:** Always double-check your derivative calculation by differentiating each term separately to avoid simple errors.

Learn how to find the derivative f'(1) for the function f(x) = 5x^4 - 12x^2 - 3x + 1. This problem involves differentiating a polynomial function and evaluating it at a specific point. Suitable for students in Grades 11-12.

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