Question
Find \( f^{\prime}(x) \) for the following function. Then find \( f^{\prime}(3), f^{\prime}(0) \), and \( f^{\prime}(-2) \). \( f(x)=7 x^{2}+4 x \) \( f^{\prime}(x)=\square \)
Ask by Smith Lang. in the United States
Feb 17,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( f^{\prime}(x) = 14x + 4 \)
\( f^{\prime}(3) = 46 \)
\( f^{\prime}(0) = 4 \)
\( f^{\prime}(-2) = -24 \)
Solution
To find the derivative of the function f(x) = 7x² + 4x, we differentiate term by term.
1. The derivative of 7x² is 14x.
2. The derivative of 4x is 4.
Thus, the derivative is:
f′(x) = 14x + 4
Now, plugging in the particular values:
• f′(3) = 14(3) + 4 = 42 + 4 = 46
• f′(0) = 14(0) + 4 = 0 + 4 = 4
• f′(–2) = 14(–2) + 4 = –28 + 4 = –24
So, the results are:
f′(x) = 14x + 4
f′(3) = 46
f′(0) = 4
f′(–2) = –24
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Simplify this solution Beyond the Answer
To find the derivative \( f^{\prime}(x) \) of the function \( f(x) = 7x^{2} + 4x \), we apply the power rule. The derivative of \( x^n \) is \( n \cdot x^{n-1} \). Thus, \[ f^{\prime}(x) = 14x + 4. \] Now evaluating the derivative at the specified points: 1. \( f^{\prime}(3) = 14(3) + 4 = 42 + 4 = 46 \). 2. \( f^{\prime}(0) = 14(0) + 4 = 0 + 4 = 4 \). 3. \( f^{\prime}(-2) = 14(-2) + 4 = -28 + 4 = -24 \). So, the answers are: - \( f^{\prime}(3) = 46 \) - \( f^{\prime}(0) = 4 \) - \( f^{\prime}(-2) = -24 \) Final results: \[ f^{\prime}(x) = 14x + 4 \] \[ f^{\prime}(3) = 46, \quad f^{\prime}(0) = 4, \quad f^{\prime}(-2) = -24 \]
