Question textConsider the following function.The derivative of a function at a point dom can be calculated using limits as follows: Determine the derivative of the function .Input your response in the box below. Enter fractions if required, do not use not decimals. Answer 1 Question 1[input]多项填空题
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Question textConsider the following function.The derivative of a function at a point dom can be calculated using limits as follows: Determine the derivative of the function .Input your response in the box below. Enter fractions if required, do not use not decimals. Answer 1 Question 1[input]多项填空题

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BlackTom AI · Accepted Answer

Using the limit definition of the derivative for f(x) = 3x^2, compute the difference quotient and take the limit as Δx → 0.

1. Relevant knowledge points and concepts
- Definition: f'(x) = lim_{Δx→0} [f(x+Δx) − f(x)] / Δx.
- Algebraic expansion of (x + Δx)^2 = x^2 + 2xΔx + (Δx)^2.
- Constant multiple and power rules: derivative of c·g(x) is c·g'(x); derivative of x^2 is 2x.

2. Complete reasoning process step by step
- Start with f(x+Δx) and compute the difference f(x+Δx) − f(x).
- Divide by Δx to form the difference quotient.
- Simplify algebraically and take the limit Δx → 0.

3. Calculations with intermediate steps
- f(x) = 3x^2.
- f(x + Δx) = 3(x + Δx)^2 = 3(x^2 + 2xΔx + Δx^2) = 3x^2 + 6xΔx + 3Δx^2.
- f(x + Δx) − f(x) = (3x^2 + 6xΔx + 3Δx^2) − 3x^2 = 6xΔx + 3Δx^2.
- Difference quotient: (6xΔx + 3Δx^2) / Δx = 6x + 3Δx.
- Take the limit: lim_{Δx→0} (6x + 3Δx) = 6x + 3·0 = 6x.

4. Why the correct answer is correct
The algebraic cancellation removes Δx from each term, leaving a linear expression in x plus a term proportional to Δx that vanishes in the limit. Alternatively, by the power rule: d/dx(3x^2) = 3·(2x) = 6x, matching the limit result.

5. Brief note on incorrect alternatives
- 3x^2 would be the original function, not its derivative.
- 2x is the derivative of x^2, but missing the constant factor 3.
- Any constant term remaining (e.g., +3) would come from algebra error; the correct limit eliminates Δx terms.

6. Conclusion confirming final answer
Therefore the derivative f'(x) = 6x, which matches the provided answer.

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