Question

If \(y = \frac{1}{3x+5}\), then the value of \(\frac{d^9y}{dx^9}\) is..

• A. \(\frac{9!\times 3^9}{(3x+5)^9}\)
• B. \(\frac{(-1)^8\times 9!\times 3^9}{(3x+5)^9}\)
• C. \(\frac{(-1)^9\times 9!\times 3^9}{(3x+5)^{10}}\)
• D. \(\frac{(-1)^8\times 8!\times 3^9}{(3x+5)^{10}}\)

Hint:

Reduce the angle 19 pi by 4, then use tan of 3 pi by 8.

Answer 1

The Correct Option is C

Step 1: Half angle formula:
$\tan\frac{\pi}{8} = \sqrt2 - 1$, so $\tan\frac{3\pi}{8} = \cot\frac{\pi}{8} = \frac{1}{\sqrt2 - 1} = \sqrt2 + 1$.

Step 2: Finish:
The argument equals $-\tan\frac{3\pi}{8}$ and $\frac{3\pi}{8}$ is in the principal range, so the value is $-\frac{3\pi}{8}$ (C).

Final Answer:
$-\frac{3\pi}{8}$. \[ \boxed{-\frac{3\pi}{8}} \]