Question
If \( f(x)=(4 x+8)^{-2} \), find \( f^{\prime}(x) \). Find \( f^{\prime}(4) \).
Ask by Spencer Vaughan. in the United States
Feb 24,2025
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Answer
The derivative of \( f(x) = (4x + 8)^{-2} \) is \( f'(x) = -8 / (4x + 8)^3 \). When \( x = 4 \), \( f'(4) = -1/1728 \).
Solution
We start with the function
f(x) = (4x + 8)^(–2).
To differentiate f(x), use the chain rule. Let u = 4x + 8, so that f(x) = u^(–2). The derivative with respect to u is
d/du [u^(–2)] = –2u^(–3).
Since du/dx = 4, applying the chain rule gives
f '(x) = –2u^(–3) ⋅ 4 = –8(4x + 8)^(–3).
This can be written as
f '(x) = –8 / (4x + 8)^3.
To find f '(4), substitute x = 4:
4(4) + 8 = 16 + 8 = 24,
so
f '(4) = –8 / (24)^3.
Since (24)^3 = 13,824, this simplifies to
f '(4) = –8 / 13,824 = –1 / 1,728.
Thus, the answers are:
f '(x) = –8 / (4x + 8)^3 and f '(4) = –1/1728.
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Simplify this solution Bonus Knowledge
To find \( f^{\prime}(x) \) for the function \( f(x)=(4x+8)^{-2} \), we will use the chain rule. 1. We differentiate \( f(x) \): \[ f^{\prime}(x) = -2(4x+8)^{-3} \cdot (4) = -8(4x+8)^{-3}. \] Now, we substitute \( x = 4 \) into \( f^{\prime}(x) \) to find \( f^{\prime}(4) \): 2. Calculate \( f(4) \): \[ f(4) = (4 \cdot 4 + 8)^{-2} = (16 + 8)^{-2} = 24^{-2} = \frac{1}{576}. \] 3. Substitute \( x = 4 \): \[ f^{\prime}(4) = -8(4 \cdot 4 + 8)^{-3} = -8 \cdot 24^{-3} = -8 \cdot \frac{1}{13824} = -\frac{8}{13824} = -\frac{1}{1728}. \] Thus, the derivative at \( x = 4 \) is: \[ f^{\prime}(4) = -\frac{1}{1728}. \]
