derivative

1. Derivative of ln(-x)

Can someone please explain to me why the derivative of ln(-x) is 1/x? Thanks!
2. The n-th derivative of x*cos(2x)

Hey, I have to find the f^{(n)}(x) of function f(x) = x \cdot \cos(2x). I know I need to find a Leibnitz formula for derivative of product, but how? Also, I have to solve this formula explicitly, so, f^{(n)}(x) = .... Please help ;)
3. Calculating derivative using formal definition tricky

Can someone guide me in terms of finishing off the derivative computation that I have started below? Help is greatly appreciated!
4. Calculate derivative using formal definition

I'm really hoping someone can help me calculate the derivative (using the formal definition) for the following function: Here is my attempted solutions but I know it is wrong: Can someone tell me what I am doing incorrectly?
5. Using formal definition of a limit

Can someone help me complete the derivative computation below? I'm a bit stuck.
6. Creating generic formula

Hi, I'm trying to create a formula for the function $g^n(x)$ which has the following derivative pattern: $$g^1(x) = \frac{e^x(x-1) + 1}{x^2}$$ $$g^2(x) = \frac{e^x(x^2 - 2x + 2) - 2}{x^4}$$ $$g^3(x) = \frac{e^x(x^3 - 3x^2 + 6x - 6) + 6}{x^4}$$ g^4(x) = \frac{e^x(x^4 - 4x^3 +12x^2 -24x + 24)...