How Do You Differentiate Functions Involving Natural Logarithms?

[jesuslovesu]

[SOLVED] need some help differentiating

Uh my bad, forgot how to differentiate ln's

Homework Statement

$$\frac{V_0}{xln(\frac{b}{x})}$$
Find dE/dx

I can almost get the answer, but I had to use MATLAB to find the actual answer, so I am kind of feeling stupid now.

My problem is when I try to differentiate the ln

Homework Equations

The Attempt at a Solution

$$d/dx(\frac{V_0}{xln(\frac{b}{x})}) = \frac{-V_0}{x^2 ln(b/x)} + \frac{V_0 * b/x^2}{x ln(b/x)^2} =$$

$$\frac{-V_0}{x^2 ln(b/x)} + \frac{V_0 * b}{x^3 ln(b/x)^2}$$

The second term should only have x^2 in it (and no b), does anyone see where I went wrong?

 

[sutupidmath]

$$log_a(x)=y=>a^{y}=x$$ let's differentiate this implicitly, we get:

$$\frac{d}{dx}(a^{y}=x)=>\frac{dy}{dx}a^{y}lna=1=>\frac{dy}{dx}=\frac{1}{a^{y}lna}=>\frac{dy}{dx}=\frac{1}{xlna}$$

When a=e, we get:$$\frac{dy}{dx}=\frac{1}{x}$$. So it is just a special case of a general case.