| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | To find $p'(2)$ for $p(x) = f(x)g(x)$, we can use the product rule for derivatives. The product rule states that if you have two functions $f(x)$ and $g(x)$, then the derivative of their product $p(x) = f(x)g(x)$ is given by: \[p'(x) = f(x)g'(x) + f'(x)g(x)\] Given: - $f(2) = 3$ - $f'(2) = -4$ - $g(2) = 1$ - $g'(2) = 6$ We substitute these values into the product rule formula to find $p'(2)$: \[p'(2) = f(2)g'(2) + f'(2)g(2)\] \[p'(2) = (3)(6) + (-4)(1)\] \[p'(2) = 18 - 4\] \[p'(2) = 14\] Therefore, $p'(2) = 14$. |
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| deepseekr1 | To find \( p'(2) \) where \( p(x) = f(x)g(x) \), we use the product rule: \[ p'(x) = f'(x)g(x) + f(x)g'(x) \] Substituting the given values at \( x = 2 \): \[ p'(2) = f'(2)g(2) + f(2)g'(2) = (-4)(1) + (3)(6) = -4 + 18 = 14 \] **Answer:** \(\boxed{14}\) |
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| gpto1 | To find \( p'(2) \), we start by differentiating \( p(x) = f(x)g(x) \) using the product rule: \[ p'(x) = f'(x)g(x) + f(x)g'(x). \] Now, we substitute the given values at \( x = 2 \): \[ \begin{align*} p'(2) &= f'(2)g(2) + f(2)g'(2) \\ &= (-4)(1) + (3)(6) \\ &= -4 + 18 \\ &= 14. \end{align*} \] **Answer:** \(14\) |
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| GPT-4o | To find \( p'(x) \) for the given function \( p(x) = f(x)g(x) \), we need to use the product rule of differentiation. The product rule states that for \( p(x) = f(x)g(x) \), the derivative \( p'(x) \) is given by: \[ p'(x) = f'(x)g(x) + f(x)g'(x) \] We are given the following values at \( x = 2 \): \[ f(2) = 3, \quad f'(2) = -4, \quad g(2) = 1, \quad g'(2) = 6 \] We need to find \( p'(2) \). Using the product rule, we have: \[ p'(2) = f'(2)g(2) + f(2)g'(2) \] Substituting the given values into the formula: \[ p'(2) = (-4)(1) + (3)(6) \] \[ p'(2) = -4 + 18 \] \[ p'(2) = 14 \] So, the value of \( p'(2) \) is: \[ \boxed{14} \] |
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| llama3.1-70B-Instruct | ## Step 1: Recall the product rule for differentiation The product rule states that if we have a function of the form $p(x) = f(x)g(x)$, then the derivative $p'(x)$ is given by $p'(x) = f'(x)g(x) + f(x)g'(x)$. ## Step 2: Apply the product rule to the given function We are given $p(x) = f(x)g(x)$ and we need to find $p'(2)$. According to the product rule, $p'(x) = f'(x)g(x) + f(x)g'(x)$. Therefore, $p'(2) = f'(2)g(2) + f(2)g'(2)$. ## Step 3: Substitute the given values into the equation We are given that $f(2) = 3$, $f'(2) = -4$, $g(2) = 1$, and $g'(2) = 6$. Substituting these values into the equation from step 2, we get $p'(2) = (-4)(1) + (3)(6)$. ## Step 4: Calculate the value of $p'(2)$ Now, we perform the arithmetic: $p'(2) = -4 + 18$. ## Step 5: Simplify to find the final answer Simplifying the expression gives us $p'(2) = 14$. The final answer is: $\boxed{14}$ |
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| llama3.1-405B-Instruct | Using the product rule, we have $$p'(x)=f'(x)g(x)+f(x)g'(x).$$Plugging in $x=2$, we have $$p'(2)=f'(2)g(2)+f(2)g'(2)=(-4)(1)+(3)(6)=\boxed{14}.$$ Final Answer: The final answer is 14. I hope it is correct. |
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| DeepSeek-Coder-V2 | To find \( p'(2) \) for the given problem, we need to use the product rule for differentiation. The product rule states that if \( p(x) = f(x)g(x) \), then the derivative \( p'(x) \) is given by: \[ p'(x) = f'(x)g(x) + f(x)g'(x) \] Given the values: - \( f(2) = 3 \) - \( f'(2) = -4 \) - \( g(2) = 1 \) - \( g'(2) = 6 \) We need to evaluate \( p'(2) \). Substituting the given values into the product rule formula, we get: \[ p'(2) = f'(2)g(2) + f(2)g'(2) \] Now, substitute the specific values: \[ p'(2) = (-4)(1) + (3)(6) \] Calculate each term: \[ p'(2) = -4 + 18 \] Finally, sum the results: \[ p'(2) = 14 \] Thus, the value of \( p'(2) \) is: \[ \boxed{14} \] |
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| Qwen2-72B-Instruct | To find $p'(2)$ for $p(x) = f(x)g(x)$, we can use the product rule for derivatives. The product rule states that if you have two functions $f(x)$ and $g(x)$, then the derivative of their product $p(x) = f(x)g(x)$ is given by: \[p'(x) = f(x)g'(x) + f'(x)g(x)\] Given: - $f(2) = 3$ - $f'(2) = -4$ - $g(2) = 1$ - $g'(2) = 6$ We substitute these values into the product rule formula to find $p'(2)$: \[p'(2) = f(2)g'(2) + f'(2)g(2)\] \[p'(2) = (3)(6) + (-4)(1)\] \[p'(2) = 18 - 4\] \[p'(2) = 14\] Therefore, $p'(2) = 14$. |
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| baichuan4 | To find $p'(2)$, we can use the product rule for derivatives, which states that for two functions $f(x)$ and $g(x)$, the derivative of their product $p(x) = f(x)g(x)$ is: $$p'(x) = f'(x)g(x) + f(x)g'(x)$$ Now, we want to evaluate $p'(2)$, so we will substitute the given values at $x = 2$ into the equation above: $$p'(2) = f'(2)g(2) + f(2)g'(2)$$ Substitute the given values: $$p'(2) = (-4)(1) + (3)(6)$$ Calculate the result: $$p'(2) = -4 + 18$$ $$p'(2) = 14$$ Therefore, the value of $p'(2)$ is $14$. |
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| doubao | We use the product rule which states that (fg)' = f'g + fg'. So, p'(2) = f'(2)g(2) + f(2)g'(2) = (-4)×1 + 3×6 = -4 + 18 = 14. |
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| minimax | To find the derivative of the product of two functions, we use the product rule. The product rule states that if you have two functions $f(x)$ and $g(x)$, then the derivative of their product $p(x) = f(x)g(x)$ is given by: $$ p'(x) = f'(x)g(x) + f(x)g'(x) $$ Given the values at $x = 2$, we can substitute them into the product rule to find $p'(2)$: $$ p'(2) = f'(2)g(2) + f(2)g'(2) $$ Now, plug in the given values: $$ p'(2) = (-4)(1) + (3)(6) $$ $$ p'(2) = -4 + 18 $$ $$ p'(2) = 14 $$ Therefore, the derivative of the product $p(x)$ at $x = 2$ is $p'(2) = 14$. |
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| kimi | To find the derivative of \( p(x) = f(x)g(x) \) at \( x = 2 \), we can use the product rule for differentiation. The product rule states that if we have two functions \( u(x) \) and \( v(x) \), then the derivative of their product \( u(x)v(x) \) is given by: \[ (uv)' = u'v + uv' \] In this problem, \( u(x) = f(x) \) and \( v(x) = g(x) \). So, we can apply the product rule to find \( p'(x) \): \[ p'(x) = f'(x)g(x) + f(x)g'(x) \] Now, we substitute \( x = 2 \) and the given values: \[ p'(2) = f'(2)g(2) + f(2)g'(2) \] Given: \[ f(2) = 3 \] \[ f'(2) = -4 \] \[ g(2) = 1 \] \[ g'(2) = 6 \] Substitute these values into the equation: \[ p'(2) = (-4)(1) + (3)(6) \] \[ p'(2) = -4 + 18 \] \[ p'(2) = 14 \] So, \( p'(2) = 14 \). |
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| wenxinyiyan | 首先，我们需要使用乘积法则来计算$p'(x)$。 乘积法则告诉我们：如果$p(x) = f(x)g(x)$，那么$p'(x) = f'(x)g(x) + f(x)g'(x)$。 现在，我们已经知道$f(2)=3$，$f'(2)=-4$，$g(2)=1$，和$g'(2)=6$。 将这些值代入乘积法则，我们得到： $p'(2) = f'(2)g(2) + f(2)g'(2)$ $= (-4)\times(1) + (3)\times(6)$ $= -4 + 18$ $= 14$ 所以，$p'(2) = 14$。 |
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| zhipu | To find \( p'(2) \) for the product of two functions \( p(x) = f(x)g(x) \), we will use the product rule for differentiation. The product rule states that if you have a function that is the product of two functions, say \( u(x) \) and \( v(x) \), then the derivative of the product is given by: \[ (uv)' = u'v + uv' \] In this case, let \( u(x) = f(x) \) and \( v(x) = g(x) \). Therefore, by the product rule: \[ p'(x) = f'(x)g(x) + f(x)g'(x) \] We need to find \( p'(2) \), so we substitute \( x = 2 \) into the derivative: \[ p'(2) = f'(2)g(2) + f(2)g'(2) \] Now, we use the given values: \( f(2) = 3 \), \( f'(2) = -4 \), \( g(2) = 1 \), and \( g'(2) = 6 \). Substituting these values in: \[ p'(2) = (-4)(1) + (3)(6) \] Calculating each term separately: \[ (-4)(1) = -4 \] \[ (3)(6) = 18 \] Now add these two results together: \[ p'(2) = -4 + 18 = 14 \] So, the value of \( p'(2) \) is \( 14 \). Therefore, the solution to the problem is: \[ p'(2) = 14 \] |
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| gpto1 | To find \( p'(2) \), we start by differentiating \( p(x) = f(x)g(x) \) using the product rule: \[ p'(x) = f'(x)g(x) + f(x)g'(x). \] Now, we substitute the given values at \( x = 2 \): \[ \begin{align*} p'(2) &= f'(2)g(2) + f(2)g'(2) \\ &= (-4)(1) + (3)(6) \\ &= -4 + 18 \\ &= 14. \end{align*} \] **Answer:** \(14\) |
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| deepseekv3 | Alright, I have this problem in front of me, and I need to find the value of \( p'(2) \) given some information about the functions \( f \) and \( g \). Let's break it down step by step. ### Understanding the Problem First, let's make sure I understand what's being asked. I have a function \( p(x) \) defined as the product of two other functions, \( f(x) \) and \( g(x) \): \[ p(x) = f(x)g(x) \] I'm given specific values for these functions and their derivatives at \( x = 2 \): - \( f(2) = 3 \) - \( f'(2) = -4 \) - \( g(2) = 1 \) - \( g'(2) = 6 \) The question is asking for the value of the derivative of \( p \) at \( x = 2 \), that is, \( p'(2) \). ### Recalling Relevant Concepts To solve this, I need to recall how to find the derivative of a product of two functions. The rule for differentiating a product is called the **Product Rule**. The Product Rule states that if \( p(x) = f(x)g(x) \), then the derivative \( p'(x) \) is: \[ p'(x) = f'(x)g(x) + f(x)g'(x) \] So, to find \( p'(2) \), I'll need to apply the Product Rule at \( x = 2 \). ### Applying the Product Rule Let's write down the Product Rule again: \[ p'(x) = f'(x)g(x) + f(x)g'(x) \] Now, I need to evaluate this at \( x = 2 \): \[ p'(2) = f'(2)g(2) + f(2)g'(2) \] I have all the values I need: - \( f'(2) = -4 \) - \( g(2) = 1 \) - \( f(2) = 3 \) - \( g'(2) = 6 \) Let's plug these values into the equation. ### Calculating \( p'(2) \) Plugging in the values: \[ p'(2) = (-4)(1) + (3)(6) \] Now, let's do the multiplication: \[ (-4)(1) = -4 \] \[ (3)(6) = 18 \] Adding these together: \[ -4 + 18 = 14 \] ### Verifying the Calculation Wait a second, I got \( p'(2) = 14 \). But I want to make sure I didn't make a mistake. Let's double-check the steps. 1. **Product Rule Application**: I correctly applied the Product Rule to get \( p'(2) = f'(2)g(2) + f(2)g'(2) \). 2. **Plugging in Values**: I substituted the given values accurately: - \( f'(2) = -4 \) - \( g(2) = 1 \) - \( f(2) = 3 \) - \( g'(2) = 6 \) 3. **Multiplication**: - \( (-4)(1) = -4 \) - \( (3)(6) = 18 \) 4. **Addition**: \( -4 + 18 = 14 \) Everything seems to add up correctly. I didn't skip any steps, and the calculations are accurate. ### Final Answer After carefully applying the Product Rule and performing the necessary calculations, I conclude that: \[ p'(2) = 14 \] So, the final answer is: \[ \boxed{14} \] |
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