Question
Let \( f(x)=3 \sec (9 x) \) \( f^{\prime}(x)=\square \)
Ask by Fowler Gardner. in the United States
Feb 24,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( f'(x) = 27 \sec(9x) \tan(9x) \)
Solution
We start with the function f(x) = 3·sec(9x).
Step 1: Identify the inner function u = 9x.
Step 2: Recall that the derivative of sec(u) with respect to u is sec(u)tan(u).
Step 3: Use the chain rule. The derivative f′(x) is:
f′(x) = 3·[sec(u)tan(u)]·(du/dx)
= 3·[sec(9x)tan(9x)]·9
Step 4: Multiply the constants:
f′(x) = 27·sec(9x)tan(9x).
Thus, the derivative f′(x) is 27 sec(9x) tan(9x).
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Simplify this solution Beyond the Answer
To find the derivative \( f'(x) = 3 \sec(9x) \tan(9x) \cdot 9 = 27 \sec(9x) \tan(9x) \). Remember, the derivative of \( \sec(u) \) is \( \sec(u) \tan(u) \cdot u' \). So, we multiply by the derivative of the inside function, which is 9 in this case. Great job working through the calculus! Also, don't forget that understanding the secant and tangent functions is key when dealing with derivatives involving trigonometric functions—knowing their graphs and properties can help visualize rates of change in different scenarios!
