Question

Find the \( n \)th order derivative of \( \log x \).

Hint:

The derivatives of \( \log x \) follow a simple pattern, with each successive derivative increasing the power of \( x \) in the denominator.

Answer 1

Step 1: First derivative of \( \log x \).
The first derivative of \( \log x \) is: \[ \frac{d}{dx} (\log x) = \frac{1}{x} \]

Step 2: Second derivative of \( \log x \).
The second derivative is the derivative of \( \frac{1}{x} \): \[ \frac{d^2}{dx^2} (\log x) = -\frac{1}{x^2} \]

Step 3: Third derivative of \( \log x \).
The third derivative is the derivative of \( -\frac{1}{x^2} \): \[ \frac{d^3}{dx^3} (\log x) = \frac{2}{x^3} \]

Step 4: General form of the \( n \)th derivative.
By observing the pattern, the \( n \)th derivative of \( \log x \) is: \[ \frac{d^n}{dx^n} (\log x) = (-1)^{n-1} \frac{(n-1)!}{x^n} \]

Final Answer: The \( n \)th order derivative of \( \log x \) is: \[ \boxed{(-1)^{n-1} \frac{(n-1)!}{x^n}} \]