| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | To find the derivative of \( h(x) = (x^{-1/2} + 2x)(7 - x^{-1}) \) at \( x = 4 \), we first need to find the derivative of \( h(x) \) using the product rule. Let's denote \( u(x) = x^{-1/2} + 2x \) and \( v(x) = 7 - x^{-1} \). The product rule states that if \( h(x) = u(x)v(x) \), then \( h'(x) = u'(x)v(x) + u(x)v'(x) \). First, we need to find the derivatives of \( u(x) \) and \( v(x) \): \[ u(x) = x^{-1/2} + 2x \] \[ u'(x) = -\frac{1}{2}x^{-3/2} + 2 \] \[ v(x) = 7 - x^{-1} \] \[ v'(x) = x^{-2} \] Now, we can find \( h'(x) \): \[ h'(x) = u'(x)v(x) + u(x)v'(x) \] \[ = \left(-\frac{1}{2}x^{-3/2} + 2\right)(7 - x^{-1}) + (x^{-1/2} + 2x)(x^{-2}) \] \[ = \left(-\frac{1}{2}x^{-3/2} + 2\right)(7 - x^{-1}) + (x^{-3/2} + 2x^{-1}) \] Now, we substitute \( x = 4 \) into \( h'(x) \): \[ h'(4) = \left(-\frac{1}{2}(4)^{-3/2} + 2\right)(7 - (4)^{-1}) + ((4)^{-3/2} + 2(4)^{-1}) \] \[ = \left(-\frac{1}{2}(0.5)^{3} + 2\right)(7 - 0.25) + ((0.5)^{3} + 2(0.25)) \] \[ = \left(-\frac{1}{2}(0.125) + 2\right)(6.75) + (0.125 + 0.5) \] \[ = \left(-0.0625 + 2\right)(6.75) + 0.625 \] \[ = (1.9375)(6.75) + 0.625 \] \[ = 13.096875 + 0.625 \] \[ = 13.721875 \] Therefore, the derivative of \( h(x) \) at \( x = 4 \) is approximately \( 13.721875 \). |
13.609 |
| deepseekr1 | To find the derivative of \( h(x) = (x^{-1/2} + 2x)(7 - x^{-1}) \) at \( x = 4 \), we use the product rule. Let \( u(x) = x^{-1/2} + 2x \) and \( v(x) = 7 - x^{-1} \). The product rule states that \( h'(x) = u'(x)v(x) + u(x)v'(x) \). First, we compute the derivatives of \( u(x) \) and \( v(x) \): - \( u(x) = x^{-1/2} + 2x \) - \( u'(x) = -\frac{1}{2}x^{-3/2} + 2 \) - \( v(x) = 7 - x^{-1} \) - \( v'(x) = x^{-2} \) Next, we evaluate these functions and their derivatives at \( x = 4 \): - \( u(4) = 4^{-1/2} + 2 \cdot 4 = \frac{1}{2} + 8 = \frac{17}{2} \) - \( v(4) = 7 - 4^{-1} = 7 - \frac{1}{4} = \frac{27}{4} \) - \( u'(4) = -\frac{1}{2} \cdot 4^{-3/2} + 2 = -\frac{1}{2} \cdot \frac{1}{8} + 2 = -\frac{1}{16} + 2 = \frac{31}{16} \) - \( v'(4) = 4^{-2} = \frac{1}{16} \) Applying the product rule at \( x = 4 \): - First term: \( u'(4) \cdot v(4) = \frac{31}{16} \cdot \frac{27}{4} = \frac{837}{64} \) - Second term: \( u(4) \cdot v'(4) = \frac{17}{2} \cdot \frac{1}{16} = \frac{17}{32} = \frac{34}{64} \) Adding these terms together: \[ h'(4) = \frac{837}{64} + \frac{34}{64} = \frac{871}{64} \] Thus, the derivative of \( h(x) \) at \( x = 4 \) is \(\boxed{\dfrac{871}{64}}\). |
13.609 |
| gpto1 | To find \( h'(x) \) when \( x = 4 \) for the function \[ h(x) = \left( x^{-1/2} + 2x \right)\left( 7 - x^{-1} \right), \] we will follow these steps: 1. **Differentiate \( h(x) \) using the Product Rule.** 2. **Simplify the derivative \( h'(x) \) as necessary.** 3. **Evaluate \( h'(x) \) at \( x = 4 \).** 4. **Simplify the numerical result to obtain the final answer.** ### **Step 1: Differentiate \( h(x) \) Using the Product Rule** Let \[ u(x) = x^{-1/2} + 2x \quad \text{and} \quad v(x) = 7 - x^{-1}. \] The product rule states that \[ h'(x) = u'(x)v(x) + u(x)v'(x). \] #### **Compute \( u'(x) \):** Differentiate \( u(x) \) term by term: \[ \begin{align*} u(x) &= x^{-1/2} + 2x, \\ u'(x) &= \frac{d}{dx}\left( x^{-1/2} \right) + \frac{d}{dx}\left( 2x \right). \end{align*} \] Using the power rule \( \frac{d}{dx}\left( x^n \right) = n x^{n-1} \): \[ \frac{d}{dx}\left( x^{-1/2} \right) = -\tfrac{1}{2} x^{-1/2 - 1} = -\tfrac{1}{2} x^{-3/2}, \quad \frac{d}{dx}\left( 2x \right) = 2. \] Therefore, \[ u'(x) = -\tfrac{1}{2} x^{-3/2} + 2. \] #### **Compute \( v'(x) \):** Differentiate \( v(x) \) term by term: \[ \begin{align*} v(x) &= 7 - x^{-1}, \\ v'(x) &= 0 - \frac{d}{dx}\left( x^{-1} \right). \end{align*} \] Using the power rule: \[ \frac{d}{dx}\left( x^{-1} \right) = - x^{-1 - 1} = - x^{-2}. \] Thus, \[ v'(x) = -\left( - x^{-2} \right) = x^{-2}. \] ### **Step 2: Write the Expression for \( h'(x) \)** Apply the product rule: \[ h'(x) = u'(x)v(x) + u(x)v'(x) = \left( -\tfrac{1}{2} x^{-3/2} + 2 \right)\left( 7 - x^{-1} \right) + \left( x^{-1/2} + 2x \right)\left( x^{-2} \right). \] ### **Step 3: Evaluate \( h'(x) \) at \( x = 4 \)** First, compute the values of \( u(x) \), \( u'(x) \), \( v(x) \), and \( v'(x) \) at \( x = 4 \): #### **Compute \( x \) Raised to Various Powers at \( x = 4 \):** - \( x^{-1/2} = 4^{-1/2} = \frac{1}{\sqrt{4}} = \frac{1}{2} \) - \( x^{-1} = 4^{-1} = \frac{1}{4} \) - \( x^{-3/2} = 4^{-3/2} = \frac{1}{4^{3/2}} = \frac{1}{8} \) - \( x^{-2} = 4^{-2} = \frac{1}{16} \) #### **Compute \( u(4) \) and \( u'(4) \):** - \( u(4) = x^{-1/2} + 2x = \frac{1}{2} + 2 \times 4 = \frac{1}{2} + 8 = 8.5 \) - \( u'(4) = -\tfrac{1}{2} x^{-3/2} + 2 = -\tfrac{1}{2} \times \frac{1}{8} + 2 = -\frac{1}{16} + 2 = \frac{31}{16} \) #### **Compute \( v(4) \) and \( v'(4) \):** - \( v(4) = 7 - x^{-1} = 7 - \frac{1}{4} = \frac{27}{4} \) - \( v'(4) = x^{-2} = \frac{1}{16} \) #### **Compute \( h'(4) \):** Now, plug the computed values into the expression for \( h'(x) \): \[ \begin{align*} h'(4) &= u'(4) v(4) + u(4) v'(4) \\ &= \left( \frac{31}{16} \right) \left( \frac{27}{4} \right) + \left( 8.5 \right) \left( \frac{1}{16} \right). \end{align*} \] ### **Step 4: Compute the Numerical Result** #### **Compute \( u'(4) v(4) \):** \[ \begin{align*} u'(4) v(4) &= \left( \frac{31}{16} \right) \left( \frac{27}{4} \right) = \frac{31 \times 27}{16 \times 4} = \frac{837}{64}. \end{align*} \] #### **Compute \( u(4) v'(4) \):** \[ \begin{align*} u(4) v'(4) &= 8.5 \times \frac{1}{16} = \frac{17}{2} \times \frac{1}{16} = \frac{17}{32}. \end{align*} \] #### **Add the Two Terms to Find \( h'(4) \):** \[ \begin{align*} h'(4) &= \frac{837}{64} + \frac{17}{32} = \frac{837}{64} + \frac{34}{64} = \frac{837 + 34}{64} = \frac{871}{64}. \end{align*} \] ### **Final Answer** \[ h'(4) = \frac{871}{64}. \] --- **Answer:** \(\boxed{\dfrac{871}{64}}\) |
13.609 |
| GPT-4o | To find \( h'(x) \) for the function \( h(x) = \left( x^{-1/2} + 2x \right) \left( 7 - x^{-1} \right) \) and then evaluate it at \( x = 4 \), we will use the product rule of differentiation. The product rule states that if \( h(x) = u(x) \cdot v(x) \), then \( h'(x) = u'(x)v(x) + u(x)v'(x) \). Let's rewrite the function in terms of \( u(x) \) and \( v(x) \): \[ u(x) = x^{-1/2} + 2x \] \[ v(x) = 7 - x^{-1} \] First, we need to find the derivatives \( u'(x) \) and \( v'(x) \). 1. **Find \( u'(x) \):** \[ u(x) = x^{-1/2} + 2x \] Using the power rule: \[ \frac{d}{dx} (x^{-1/2}) = -\frac{1}{2} x^{-3/2} \] \[ \frac{d}{dx} (2x) = 2 \] Thus, \[ u'(x) = -\frac{1}{2} x^{-3/2} + 2 \] 2. **Find \( v'(x) \):** \[ v(x) = 7 - x^{-1} \] Using the power rule: \[ \frac{d}{dx} (7) = 0 \] \[ \frac{d}{dx} (x^{-1}) = -x^{-2} \] Thus, \[ v'(x) = 0 + x^{-2} = x^{-2} \] Now, apply the product rule: \[ h'(x) = u'(x) v(x) + u(x) v'(x) \] Substitute \( u(x) \), \( v(x) \), \( u'(x) \), and \( v'(x) \): \[ h'(x) = \left( -\frac{1}{2} x^{-3/2} + 2 \right) (7 - x^{-1}) + \left( x^{-1/2} + 2x \right) x^{-2} \] Next, we simplify and evaluate \( h'(x) \) at \( x = 4 \): 1. **Evaluate each term at \( x = 4 \):** \[ u(4) = 4^{-1/2} + 2 \cdot 4 = \frac{1}{2} + 8 = 8.5 \] \[ u'(4) = -\frac{1}{2} \cdot 4^{-3/2} + 2 = -\frac{1}{2} \cdot \frac{1}{8} + 2 = -\frac{1}{16} + 2 = \frac{31}{16} \] \[ v(4) = 7 - 4^{-1} = 7 - \frac{1}{4} = \frac{27}{4} \] \[ v'(4) = 4^{-2} = \frac{1}{16} \] 2. **Combine the results:** \[ h'(4) = \left( \frac{31}{16} \right) \left( \frac{27}{4} \right) + \left( 8.5 \right) \left( \frac{1}{16} \right) \] \[ h'(4) = \frac{31 \cdot 27}{64} + \frac{8.5}{16} \] \[ h'(4) = \frac{837}{64} + \frac{8.5}{16} \] \[ h'(4) = \frac{837}{64} + \frac{8.5 \cdot 4}{64} \] \[ h'(4) = \frac{837}{64} + \frac{34}{64} \] \[ h'(4) = \frac{871}{64} \] \[ h'(4) = 13.609375 \] Therefore, \( h'(x) \) at \( x = 4 \) is \( 13.609375 \). |
13.609 |
| llama3.1-70B-Instruct | To find h'(x) when x = 4, we need to find the derivative of h(x) and then evaluate it at x = 4. First, let's find the derivative of h(x): h(x) = (x^{-1/2} + 2x)(7 - x^{-1}) Using the product rule, we get: h'(x) = (x^{-1/2} + 2x)'(7 - x^{-1}) + (x^{-1/2} + 2x)(7 - x^{-1})' = (-1/2x^{-3/2} + 2)(7 - x^{-1}) + (x^{-1/2} + 2x)(x^{-2}) Simplifying, we get: h'(x) = (-1/2x^{-3/2} + 2)(7 - x^{-1}) + x^{-3/2} + 2x^{-1} Now, let's evaluate h'(x) at x = 4: h'(4) = (-1/2(4)^{-3/2} + 2)(7 - (4)^{-1}) + (4)^{-3/2} + 2(4)^{-1} = (-1/2(1/8) + 2)(7 - 1/4) + 1/8 + 1/2 = (-1/16 + 2)(27/4) + 1/8 + 1/2 = (31/16)(27/4) + 1/8 + 1/2 = 837/64 + 1/8 + 1/2 = 837/64 + 8/64 + 32/64 = 877/64 Therefore, h'(4) = 877/64. |
13.609 |
| llama3.1-405B-Instruct | To find the derivative of h(x), we can use the product rule and the chain rule. First, let's rewrite h(x) as: h(x) = (x^{-1/2} + 2x)(7 - x^{-1}) = x^{-1/2}(7 - x^{-1}) + 2x(7 - x^{-1}) Now, we can take the derivative of each term separately: h'(x) = d/dx (x^{-1/2}(7 - x^{-1})) + d/dx (2x(7 - x^{-1})) Using the product rule, we get: h'(x) = x^{-1/2} d/dx (7 - x^{-1}) + (7 - x^{-1}) d/dx (x^{-1/2}) + 2x d/dx (7 - x^{-1}) + (7 - x^{-1}) d/dx (2x) Simplifying and using the chain rule, we get: h'(x) = x^{-1/2} (x^{-2}) + (7 - x^{-1}) (-1/2 x^{-3/2}) + 2x (-x^{-2}) + (7 - x^{-1}) (2) Combining like terms, we get: h'(x) = x^{-5/2} - 1/2 x^{-3/2} (7 - x^{-1}) - 2x^{-1} + 14 - 2x^{-1} Now, we can evaluate h'(x) at x = 4: h'(4) = 4^{-5/2} - 1/2 (4^{-3/2}) (7 - 4^{-1}) - 2(4^{-1}) + 14 - 2(4^{-1}) Simplifying, we get: h'(4) = 1/32 - 1/2 (1/8) (7 - 1/4) - 1/2 + 14 - 1/2 = 1/32 - 1/16 (27/4) - 1/2 + 14 - 1/2 = 1/32 - 27/64 - 1/2 + 14 - 1/2 = 1/32 - 27/64 - 1 + 14 = 1/32 - 27/64 + 13 = (64 - 27 + 832)/64 = 869/64 Therefore, h'(4) = 869/64. |
13.609 |
| DeepSeek-Coder-V2 | To solve the problem of finding h'(x) when x = 4 for the function h(x) = (x^{-1/2} + 2x)(7 - x^{-1}), we will follow these steps: 1. **Expand the function h(x)**: \[ h(x) = (x^{-1/2} + 2x)(7 - x^{-1}) \] Distribute the terms inside the parentheses: \[ h(x) = x^{-1/2}(7 - x^{-1}) + 2x(7 - x^{-1}) \] \[ h(x) = 7x^{-1/2} - x^{-3/2} + 14x - 2x^0 \] Simplify the terms: \[ h(x) = 7x^{-1/2} - x^{-3/2} + 14x - 2 \] 2. **Find the derivative h'(x) using the power rule**: \[ h(x) = 7x^{-1/2} - x^{-3/2} + 14x - 2 \] Differentiate each term separately: \[ \frac{d}{dx}(7x^{-1/2}) = 7 \cdot \left(-\frac{1}{2}\right) x^{-3/2} = -\frac{7}{2} x^{-3/2} \] \[ \frac{d}{dx}(-x^{-3/2}) = -1 \cdot \left(-\frac{3}{2}\right) x^{-5/2} = \frac{3}{2} x^{-5/2} \] \[ \frac{d}{dx}(14x) = 14 \] \[ \frac{d}{dx}(-2) = 0 \] Combine the derivatives: \[ h'(x) = -\frac{7}{2} x^{-3/2} + \frac{3}{2} x^{-5/2} + 14 \] 3. **Evaluate h'(x) at x = 4**: \[ h'(4) = -\frac{7}{2} (4)^{-3/2} + \frac{3}{2} (4)^{-5/2} + 14 \] Calculate the powers of 4: \[ 4^{-3/2} = \left(4^{1/2}\right)^{-3} = 2^{-3} = \frac{1}{8} \] \[ 4^{-5/2} = \left(4^{1/2}\right)^{-5} = 2^{-5} = \frac{1}{32} \] Substitute these values back into the expression: \[ h'(4) = -\frac{7}{2} \cdot \frac{1}{8} + \frac{3}{2} \cdot \frac{1}{32} + 14 \] Simplify the fractions: \[ h'(4) = -\frac{7}{16} + \frac{3}{64} + 14 \] Convert \(\frac{7}{16}\) to a common denominator of 64: \[ -\frac{7}{16} = -\frac{28}{64} \] Combine the fractions: \[ h'(4) = -\frac{28}{64} + \frac{3}{64} + 14 = -\frac{25}{64} + 14 \] Convert 14 to a fraction with a denominator of 64: \[ 14 = \frac{896}{64} \] Add the fractions: \[ h'(4) = -\frac{25}{64} + \frac{896}{64} = \frac{871}{64} \] Thus, the value of h'(x) when x = 4 is: \[ h'(4) = \frac{871}{64} \] |
13.609 |
| Qwen2-72B-Instruct | To find the derivative of h(x) = (x^{-1/2} + 2x)(7 - x^{-1}) at x = 4, we first need to find the derivative of h(x) using the product rule. Let's denote u(x) = x^{-1/2} + 2x and v(x) = 7 - x^{-1}. The product rule states that if h(x) = u(x)v(x), then h'(x) = u'(x)v(x) + u(x)v'(x). First, we need to find the derivatives of u(x) and v(x): u(x) = x^{-1/2} + 2x u'(x) = -\frac{1}{2}x^{-3/2} + 2 v(x) = 7 - x^{-1} v'(x) = x^{-2} Now, we can find h'(x): h'(x) = u'(x)v(x) + u(x)v'(x) = \left(-\frac{1}{2}x^{-3/2} + 2\right)(7 - x^{-1}) + (x^{-1/2} + 2x)(x^{-2}) = \left(-\frac{1}{2}x^{-3/2} + 2\right)(7 - x^{-1}) + (x^{-3/2} + 2x^{-1}) Now, we substitute x = 4 into h'(x): h'(4) = \left(-\frac{1}{2}(4)^{-3/2} + 2\right)(7 - (4)^{-1}) + ((4)^{-3/2} + 2(4)^{-1}) = \left(-\frac{1}{2}(0.5)^{3} + 2\right)(7 - 0.25) + ((0.5)^{3} + 2(0.25)) = \left(-\frac{1}{2}(0.125) + 2\right)(6.75) + (0.125 + 0.5) = \left(-0.0625 + 2\right)(6.75) + 0.625 = (1.9375)(6.75) + 0.625 = 13.096875 + 0.625 = 13.721875 Therefore, the derivative of h(x) at x = 4 is approximately 13.721875. |
13.609 |
| baichuan4 | To find the derivative of the function \(h(x) = (x^{-1/2} + 2x)(7 - x^{-1})\), we can use the product rule, which states that for two functions \(f(x)\) and \(g(x)\), their derivative \((fg)'\) is \(f'g + fg'\). First, let's identify \(f(x)\) and \(g(x)\): - \(f(x) = x^{-1/2} + 2x\) - \(g(x) = 7 - x^{-1}\) Now, we need to find \(f'(x)\) and \(g'(x)\): - \(f'(x) = -\frac{1}{2}x^{-3/2} + 2\) - \(g'(x) = 0 - (-x^{-2}) = x^{-2}\) Applying the product rule, we get: \[h'(x) = f'(x)g(x) + f(x)g'(x)\] \[h'(x) = (-\frac{1}{2}x^{-3/2} + 2)(7 - x^{-1}) + (x^{-1/2} + 2x)x^{-2}\] Now, we can simplify this expression: \[h'(x) = (-\frac{7}{2}x^{-3/2} + 14 - x^{-1} + 2x^{-1/2} - 2x^{-1}) + (x^{-3/2} + 2 - 2x^{-1})\] \[h'(x) = -\frac{7}{2}x^{-3/2} + 14 - x^{-1} + 2x^{-1/2} - 2x^{-1} + x^{-3/2} + 2 - 2x^{-1}\] \[h'(x) = -\frac{5}{2}x^{-3/2} + 16 - 5x^{-1} + 2x^{-1/2}\] Now, we need to find \(h'(4)\): \[h'(4) = -\frac{5}{2}(4)^{-3/2} + 16 - 5(4)^{-1} + 2(4)^{-1/2}\] \[h'(4) = -\frac{5}{2}(\frac{1}{8}) + 16 - \frac{5}{4} + 2(\frac{1}{2})\] \[h'(4) = -\frac{5}{16} + 16 - \frac{5}{4} + 1\] \[h'(4) = 16 + 1 - \frac{5}{4} - \frac{5}{16}\] \[h'(4) = 17 - \frac{20}{16} - \frac{5}{16}\] \[h'(4) = 17 - \frac{25}{16}\] \[h'(4) = \frac{272}{16} - \frac{25}{16}\] \[h'(4) = \frac{247}{16}\] So, \(h'(4) = \frac{247}{16}\). |
13.609 |
| doubao | First find the derivative of h(x) using the product rule and the chain rule. Let f(x) = x^{-1/2} + 2x and g(x) = 7 - x^{-1}. f'(x) = -1/2x^{-3/2} + 2. g'(x) = x^{-2}. h'(x) = f'(x)g(x) + f(x)g'(x) = (-1/2x^{-3/2} + 2)(7 - x^{-1}) + (x^{-1/2} + 2x)x^{-2}. Now evaluate h'(4): h'(4) = (-1/2(4)^{-3/2} + 2)(7 - (4)^{-1}) + ((4)^{-1/2} + 2(4))(4)^{-2} = (-1/2(1/8) + 2)(7 - 1/4) + ((1/2) + 8)(1/16) = (-1/16 + 2)(27/4) + (17/2)(1/16) = (31/16)(27/4) + 17/32 = 837/64 + 17/32 = 837/64 + 34/64 = 871/64. |
13.609 |
| minimax | To find the derivative of the function h(x) at x = 4, we first need to differentiate h(x) with respect to x. The function h(x) is given by: h(x) = (x^{-1/2} + 2x)(7 - x^{-1}) We can use the product rule to differentiate this function. The product rule states that if you 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' Let's define u(x) and v(x) as follows: u(x) = x^{-1/2} + 2x v(x) = 7 - x^{-1} Now we find the derivatives of u(x) and v(x) separately: u'(x) = d/dx (x^{-1/2}) + d/dx (2x) = -1/2 * x^{-3/2} + 2 v'(x) = d/dx (7) - d/dx (x^{-1}) = 0 + 1 * x^{-2} = x^{-2} Now we apply the product rule: h'(x) = u'(x)v(x) + u(x)v'(x) Substitute the expressions for u(x), v(x), u'(x), and v'(x): h'(x) = (-1/2 * x^{-3/2} + 2)(7 - x^{-1}) + (x^{-1/2} + 2x)(x^{-2}) Now we need to simplify this expression: h'(x) = (-1/2 * x^{-3/2} * 7 + 2 * 7 - 1/2 * x^{-3/2} * x^{-1} + 2 * x^{-1}) + (x^{-1/2} * x^{-2} + 2x * x^{-2}) Simplify further: h'(x) = (-7/2 * x^{-3/2} + 14 - 1/2 * x^{-5/2} + 2 * x^{-1}) + (x^{-5/2} + 2 * x^{-1}) Combine like terms: h'(x) = (-7/2 * x^{-3/2} - 1/2 * x^{-5/2}) + (14 + 2 * x^{-1} + x^{-5/2} + 2 * x^{-1}) h'(x) = (-4/2 * x^{-3/2} - 1/2 * x^{-5/2}) + (14 + 4 * x^{-1}) h'(x) = -2 * x^{-3/2} - 1/2 * x^{-5/2} + 14 + 4 * x^{-1} Now we need to evaluate h'(x) at x = 4: h'(4) = -2 * 4^{-3/2} - 1/2 * 4^{-5/2} + 14 + 4 * 4^{-1} h'(4) = -2 * (1/4)^{3/2} - 1/2 * (1/4)^{5/2} + 14 + 4 * (1/4) h'(4) = -2 * (1/8) - 1/2 * (1/32) + 14 + 1 h'(4) = -1/4 - 1/64 + 14 + 1 h'(4) = -16/64 - 1/64 + 14 + 1 h'(4) = (-16 - 1)/64 + 14 + 1 h'(4) = -17/64 + 14 + 1 h'(4) = -17/64 + 96/64 + 64/64 h'(4) = (96 + 64 - 17)/64 h'(4) = 143/64 So the derivative of h(x) when x = 4 is h'(4) = 143/64. |
13.609 |
| kimi | To find the derivative of the function \( h(x) = (x^{-1/2} + 2x)(7 - x^{-1}) \), we will first expand the function using the distributive property (also known as the FOIL method for binomials) and then differentiate term by term. The function can be expanded as follows: \[ h(x) = x^{-1/2} \cdot 7 + x^{-1/2} \cdot (-x^{-1}) + 2x \cdot 7 + 2x \cdot (-x^{-1}) \] \[ h(x) = 7x^{-1/2} - x^{-3/2} + 14x - 2x^0 \] \[ h(x) = 7x^{-1/2} - x^{-3/2} + 14x - 2 \] Now, let's find the derivative \( h'(x) \) term by term: 1. The derivative of \( 7x^{-1/2} \) is \( -\frac{7}{2}x^{-3/2} \) (using the power rule \( \frac{d}{dx}x^n = nx^{n-1} \) and multiplying by the constant 7). 2. The derivative of \( -x^{-3/2} \) is \( \frac{3}{2}x^{-5/2} \) (using the power rule and multiplying by the constant -1). 3. The derivative of \( 14x \) is \( 14 \) (since the derivative of \( x \) is 1 and we multiply by the constant 14). 4. The derivative of \( -2 \) is \( 0 \) (since the derivative of a constant is 0). So, the derivative \( h'(x) \) is: \[ h'(x) = -\frac{7}{2}x^{-3/2} + \frac{3}{2}x^{-5/2} + 14 \] Now, we will evaluate \( h'(x) \) at \( x = 4 \): \[ h'(4) = -\frac{7}{2} \cdot 4^{-3/2} + \frac{3}{2} \cdot 4^{-5/2} + 14 \] \[ h'(4) = -\frac{7}{2} \cdot \frac{1}{8} + \frac{3}{2} \cdot \frac{1}{32} + 14 \] \[ h'(4) = -\frac{7}{16} + \frac{3}{64} + 14 \] To combine the fractions, we need a common denominator, which is 64: \[ h'(4) = -\frac{28}{64} + \frac{3}{64} + 14 \] \[ h'(4) = -\frac{25}{64} + 14 \] Now, we can convert 14 to a fraction with the denominator 64: \[ h'(4) = -\frac{25}{64} + \frac{896}{64} \] \[ h'(4) = \frac{871}{64} \] So, the value of \( h'(x) \) when \( x = 4 \) is \( \frac{871}{64} \). |
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| wenxinyiyan | To find h'(x) when x = 4, we first need to find the derivative of h(x) with respect to x. Given h(x) = (x^{-1/2} + 2x)(7 - x^{-1}), we can use the product rule to find its derivative. Let u = x^{-1/2} + 2x and v = 7 - x^{-1}. Then, u' = -1/2*x^{-3/2} + 2 and v' = x^{-2}. Using the product rule, h'(x) = u'v + uv'. So, h'(x) = (-1/2*x^{-3/2} + 2)(7 - x^{-1}) + (x^{-1/2} + 2x)(x^{-2}). Now, we can simplify h'(x) and then evaluate it at x = 4. After simplification, we get: h'(x) = -7/2*x^{-3/2} + 14 - 2x^{-1} + x^{-1/2} + 2x^{-1}. Now, substituting x = 4, we get: h'(4) = -7/2*(1/8) + 14 - 2*(1/4) + (1/2) + 2*(1/4). = -7/16 + 14 - 1/2 + 1/2 + 1/2. = 14 - 7/16. = 217/16. So, h'(4) = 217/16. |
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| zhipu | To find the derivative of the function \( h(x) = (x^{-1/2} + 2x)(7 - x^{-1}) \) at \( x = 4 \), we need to apply the product rule for differentiation, which states that if you have a function \( u(x)v(x) \), its derivative \( (uv)' \) is \( u'v + uv' \). Let's define: \( u(x) = x^{-1/2} + 2x \) \( v(x) = 7 - x^{-1} \) First, we find the derivatives of \( u(x) \) and \( v(x) \) with respect to \( x \). The derivative of \( u(x) \) is: \( u'(x) = \frac{d}{dx}(x^{-1/2}) + \frac{d}{dx}(2x) \) \( u'(x) = -\frac{1}{2}x^{-3/2} + 2 \) The derivative of \( v(x) \) is: \( v'(x) = \frac{d}{dx}(7) - \frac{d}{dx}(x^{-1}) \) \( v'(x) = 0 + x^{-2} \) \( v'(x) = x^{-2} \) Now, we apply the product rule: \( h'(x) = u'(x)v(x) + u(x)v'(x) \) Substitute \( u(x) \), \( u'(x) \), \( v(x) \), and \( v'(x) \) into the equation: \( h'(x) = (-\frac{1}{2}x^{-3/2} + 2)(7 - x^{-1}) + (x^{-1/2} + 2x)(x^{-2}) \) We need to evaluate this derivative at \( x = 4 \). So let's substitute \( x = 4 \) into the derivative expression: \( h'(4) = (-\frac{1}{2}(4)^{-3/2} + 2)(7 - 4^{-1}) + ((4)^{-1/2} + 2(4))(4^{-2}) \) Now calculate each part: \( 4^{-3/2} = (2^2)^{-3/2} = 2^{-3} = \frac{1}{8} \) \( 4^{-1} = \frac{1}{4} \) \( 4^{-1/2} = \frac{1}{2} \) \( 4^{-2} = \frac{1}{16} \) Substitute these values back into the equation: \( h'(4) = (-\frac{1}{2} \cdot \frac{1}{8} + 2)(7 - \frac{1}{4}) + (\frac{1}{2} + 8)(\frac{1}{16}) \) \( h'(4) = (-\frac{1}{16} + 2)(\frac{27}{4}) + (\frac{1}{2} + 8)(\frac{1}{16}) \) \( h'(4) = (\frac{31}{16})(\frac{27}{4}) + (\frac{17}{2})(\frac{1}{16}) \) Now multiply the fractions: \( h'(4) = \frac{31 \cdot 27}{16 \cdot 4} + \frac{17}{32} \) \( h'(4) = \frac{837}{64} + \frac{17}{32} \) To combine these fractions, we need a common denominator, which is 64: \( h'(4) = \frac{837}{64} + \frac{34}{64} \) \( h'(4) = \frac{837 + 34}{64} \) \( h'(4) = \frac{871}{64} \) Therefore, the value of \( h'(x) \) when \( x = 4 \) is \( \frac{871}{64} \). This can also be expressed as a mixed number if desired. To convert \( \frac{871}{64} \) to a mixed number, we perform the division of 871 by 64: 871 divided by 64 is 13 with a remainder of 43. So, \( \frac{871}{64} \) as a mixed number is \( 13 \frac{43}{64} \). The final answer is \( h'(4) = \frac{871}{64} \) or \( 13 \frac{43}{64} \). |
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| gpto1 | To find \( h'(x) \) when \( x = 4 \) for the function \[ h(x) = \left( x^{-1/2} + 2x \right)\left( 7 - x^{-1} \right), \] we will follow these steps: 1. **Differentiate \( h(x) \) using the Product Rule.** 2. **Simplify the derivative \( h'(x) \) as necessary.** 3. **Evaluate \( h'(x) \) at \( x = 4 \).** 4. **Simplify the numerical result to obtain the final answer.** ### **Step 1: Differentiate \( h(x) \) Using the Product Rule** Let \[ u(x) = x^{-1/2} + 2x \quad \text{and} \quad v(x) = 7 - x^{-1}. \] The product rule states that \[ h'(x) = u'(x)v(x) + u(x)v'(x). \] #### **Compute \( u'(x) \):** Differentiate \( u(x) \) term by term: \[ \begin{align*} u(x) &= x^{-1/2} + 2x, \\ u'(x) &= \frac{d}{dx}\left( x^{-1/2} \right) + \frac{d}{dx}\left( 2x \right). \end{align*} \] Using the power rule \( \frac{d}{dx}\left( x^n \right) = n x^{n-1} \): \[ \frac{d}{dx}\left( x^{-1/2} \right) = -\tfrac{1}{2} x^{-1/2 - 1} = -\tfrac{1}{2} x^{-3/2}, \quad \frac{d}{dx}\left( 2x \right) = 2. \] Therefore, \[ u'(x) = -\tfrac{1}{2} x^{-3/2} + 2. \] #### **Compute \( v'(x) \):** Differentiate \( v(x) \) term by term: \[ \begin{align*} v(x) &= 7 - x^{-1}, \\ v'(x) &= 0 - \frac{d}{dx}\left( x^{-1} \right). \end{align*} \] Using the power rule: \[ \frac{d}{dx}\left( x^{-1} \right) = - x^{-1 - 1} = - x^{-2}. \] Thus, \[ v'(x) = -\left( - x^{-2} \right) = x^{-2}. \] ### **Step 2: Write the Expression for \( h'(x) \)** Apply the product rule: \[ h'(x) = u'(x)v(x) + u(x)v'(x) = \left( -\tfrac{1}{2} x^{-3/2} + 2 \right)\left( 7 - x^{-1} \right) + \left( x^{-1/2} + 2x \right)\left( x^{-2} \right). \] ### **Step 3: Evaluate \( h'(x) \) at \( x = 4 \)** First, compute the values of \( u(x) \), \( u'(x) \), \( v(x) \), and \( v'(x) \) at \( x = 4 \): #### **Compute \( x \) Raised to Various Powers at \( x = 4 \):** - \( x^{-1/2} = 4^{-1/2} = \frac{1}{\sqrt{4}} = \frac{1}{2} \) - \( x^{-1} = 4^{-1} = \frac{1}{4} \) - \( x^{-3/2} = 4^{-3/2} = \frac{1}{4^{3/2}} = \frac{1}{8} \) - \( x^{-2} = 4^{-2} = \frac{1}{16} \) #### **Compute \( u(4) \) and \( u'(4) \):** - \( u(4) = x^{-1/2} + 2x = \frac{1}{2} + 2 \times 4 = \frac{1}{2} + 8 = 8.5 \) - \( u'(4) = -\tfrac{1}{2} x^{-3/2} + 2 = -\tfrac{1}{2} \times \frac{1}{8} + 2 = -\frac{1}{16} + 2 = \frac{31}{16} \) #### **Compute \( v(4) \) and \( v'(4) \):** - \( v(4) = 7 - x^{-1} = 7 - \frac{1}{4} = \frac{27}{4} \) - \( v'(4) = x^{-2} = \frac{1}{16} \) #### **Compute \( h'(4) \):** Now, plug the computed values into the expression for \( h'(x) \): \[ \begin{align*} h'(4) &= u'(4) v(4) + u(4) v'(4) \\ &= \left( \frac{31}{16} \right) \left( \frac{27}{4} \right) + \left( 8.5 \right) \left( \frac{1}{16} \right). \end{align*} \] ### **Step 4: Compute the Numerical Result** #### **Compute \( u'(4) v(4) \):** \[ \begin{align*} u'(4) v(4) &= \left( \frac{31}{16} \right) \left( \frac{27}{4} \right) = \frac{31 \times 27}{16 \times 4} = \frac{837}{64}. \end{align*} \] #### **Compute \( u(4) v'(4) \):** \[ \begin{align*} u(4) v'(4) &= 8.5 \times \frac{1}{16} = \frac{17}{2} \times \frac{1}{16} = \frac{17}{32}. \end{align*} \] #### **Add the Two Terms to Find \( h'(4) \):** \[ \begin{align*} h'(4) &= \frac{837}{64} + \frac{17}{32} = \frac{837}{64} + \frac{34}{64} = \frac{837 + 34}{64} = \frac{871}{64}. \end{align*} \] ### **Final Answer** \[ h'(4) = \frac{871}{64}. \] --- **Answer:** \(\boxed{\dfrac{871}{64}}\) |
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| deepseekv3 | To find the derivative \( h'(x) \) of the function \( h(x) = (x^{-1/2} + 2x)(7 - x^{-1}) \) and evaluate it at \( x = 4 \), we will use the **Product Rule**. The Product Rule states that if \( h(x) = f(x)g(x) \), then: \[ h'(x) = f'(x)g(x) + f(x)g'(x) \] ### Step 1: Identify and Differentiate the Components Let: \[ f(x) = x^{-1/2} + 2x \] \[ g(x) = 7 - x^{-1} \] Now compute the derivatives of \( f(x) \) and \( g(x) \): \[ f'(x) = \frac{d}{dx}(x^{-1/2}) + \frac{d}{dx}(2x) = -\frac{1}{2}x^{-3/2} + 2 \] \[ g'(x) = \frac{d}{dx}(7) - \frac{d}{dx}(x^{-1}) = 0 - (-1)x^{-2} = x^{-2} \] ### Step 2: Apply the Product Rule Using the Product Rule: \[ h'(x) = f'(x)g(x) + f(x)g'(x) \] Substitute \( f(x) \), \( f'(x) \), \( g(x) \), and \( g'(x) \): \[ h'(x) = \left(-\frac{1}{2}x^{-3/2} + 2\right)(7 - x^{-1}) + (x^{-1/2} + 2x)(x^{-2}) \] ### Step 3: Simplify Expand and simplify the expression: \[ h'(x) = \left(-\frac{1}{2}x^{-3/2} + 2\right)(7 - x^{-1}) + (x^{-1/2} + 2x)(x^{-2}) \] \[ = -\frac{1}{2}x^{-3/2} \cdot 7 + \frac{1}{2}x^{-5/2} + 2 \cdot 7 - 2x^{-1} + x^{-1/2} \cdot x^{-2} + 2x \cdot x^{-2} \] \[ = -\frac{7}{2}x^{-3/2} + \frac{1}{2}x^{-5/2} + 14 - 2x^{-1} + x^{-5/2} + 2x^{-1} \] Combine like terms: \[ h'(x) = -\frac{7}{2}x^{-3/2} + \left(\frac{1}{2}x^{-5/2} + x^{-5/2}\right) + 14 + (-2x^{-1} + 2x^{-1}) \] \[ = -\frac{7}{2}x^{-3/2} + \frac{3}{2}x^{-5/2} + 14 \] ### Step 4: Evaluate at \( x = 4 \) Substitute \( x = 4 \) into \( h'(x) \): \[ h'(4) = -\frac{7}{2}(4)^{-3/2} + \frac{3}{2}(4)^{-5/2} + 14 \] Simplify the exponents: \[ 4^{-3/2} = \frac{1}{4^{3/2}} = \frac{1}{8}, \quad 4^{-5/2} = \frac{1}{4^{5/2}} = \frac{1}{32} \] Substitute these values: \[ h'(4) = -\frac{7}{2}\left(\frac{1}{8}\right) + \frac{3}{2}\left(\frac{1}{32}\right) + 14 \] \[ = -\frac{7}{16} + \frac{3}{64} + 14 \] Combine the fractions: \[ h'(4) = -\frac{28}{64} + \frac{3}{64} + 14 \] \[ = -\frac{25}{64} + 14 \] \[ = \frac{896}{64} - \frac{25}{64} \] \[ = \frac{871}{64} \] ### Final Answer: \[ h'(4) = \frac{871}{64} \] |
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