Let [math: f(t)=t2+t3]f(t)=t^{2}+\sqrt{t^{3}}. Find [math: f′(2)]f'(2).

Question

BlackTom AI · Accepted Answer

We start by understanding the function: f(t) = t^2 + sqrt(t^3) = t^2 + t^(3/2).

Step 1: Differentiate f(t).
- The derivative of t^2 with respect to t is 2t.
- The derivative of t^(3/2) is (3/2) t^(1/2) by the power rule.
So f'(t) = 2t + (3/2) t^(1/2) = 2t + (3/2) sqrt(t).

Step 2: Evaluate f'(t) at t = 2.
- 2t at t=2 is 2*2 = 4.
- (3/2) sqrt(t) at t=2 is (3/2) * sqrt(2).
Thus f'(2) = 4 + (3/2) sqrt(2).

Step 3: Compare with the given options.
- Option a: It presents a structure that appears to combine extraneous numbers with a fraction involving 3/(2√2) and an added 2; this does not simplify to 4 + (3/2)√2. The numerical form seems inconsistent with the derived expression.
- Option b: Its expression includes (322+4) multiplied by (3√2)/2, plus 4. This is not equivalent to 4 + (3/2)√2 and introduces an incorrect large coefficient (322+4).
- Option c: It shows (322+4) times (3)/(2√2) plus 4. Again, this does not match the correct form 4 + (3/2)√2 and uses an odd coefficient structure.
- Option d: It has (322+2) times (3√2)/2 plus 2. This is also not equivalent to 4 + (3/2)√2 and misuses the constants and additions.

Step 4: Summarize why the options do not align with the correct derivative value.
- The correct expression for f'(2) derived from standard differentiation is 4 + (3/2)√2, which is a simple sum of 4 and a multiple of √2.
- Each option a–d combines numbers like 322, 2, 4, and multipliers (3)/(2√2) or (3√2)/2 in ways that do not simplify to 4 + (3/2)√2. The presence of 322 in several options indicates a mis-transcription or formatting error, and the overall structure does not reduce to the computed value.

Conclusion (without stating it explicitly): Therefore, none of the presented options reproduce the exact result 4 + (3/2)√2; at least one of them contains arithmetic or symbolic inconsistencies that prevent equality with the correct derivative at t = 2.

Let [math: f(t)=t2+t3]f(t)=t^{2}+\sqrt{t^{3}}. Find [math: f′(2)]f'(2).单项选择题

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