Pregunta
If \( f(x)=3 \sec (6 x) \), find \( f^{\prime}(x) \). Find \( f^{\prime}(1) \).
Ask by Mcdonald Ortega. in the United States
Feb 25,2025
Solución de inteligencia artificial de Upstudy
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\( f^{\prime}(x) = 18 \sec(6x) \tan(6x) \)
\( f^{\prime}(1) = 18 \sec(6) \tan(6) \)
Solución
We are given f(x) = 3 sec(6x). To find the derivative f′(x), we first recall the derivative of sec(u) with respect to x:
d/dx [sec(u)] = sec(u) tan(u) · u′.
Here, u = 6x, so u′ = 6. Therefore,
f′(x) = 3 · [d/dx sec(6x)]
= 3 · [sec(6x) tan(6x) · 6]
= 18 sec(6x) tan(6x).
Next, to find f′(1), substitute x = 1 into the expression:
f′(1) = 18 sec(6 · 1) tan(6 · 1)
= 18 sec(6) tan(6).
Thus, the derivative and its value at x = 1 are:
f′(x) = 18 sec(6x) tan(6x)
f′(1) = 18 sec(6) tan(6).
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Simplifique esta solución The Deep Dive
To find the derivative of the function \( f(x) = 3 \sec(6x) \), we will use the chain rule. The derivative of \( \sec(u) \) is \( \sec(u) \tan(u) \cdot u' \). First, we identify \( u = 6x \), which means \( u' = 6 \). Therefore, we have: \[ f'(x) = 3 \cdot \sec(6x) \tan(6x) \cdot 6 = 18 \sec(6x) \tan(6x). \] Now, we need to find \( f'(1) \): \[ f'(1) = 18 \sec(6 \cdot 1) \tan(6 \cdot 1) = 18 \sec(6) \tan(6). \] Thus, the final answer is: \[ f'(1) = 18 \sec(6) \tan(6). \]
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