# Lny and derivatives

1. Dec 14, 2008

### sonofjohn

Why is the lny y'\y?

could I also say the lny is dy\y?

2. Dec 14, 2008

### Staff: Mentor

Re: lny

Neither of these is correct.
The derivative with respect to y of ln y is 1/y. In symbols, d/dy(ln y) = 1/y
The differential of ln y, d(ln y), is dy/y.

Is that clear?

3. Dec 14, 2008

### sonofjohn

Re: lny

I do understand what I should have said is when taking the derivative with respect to x, why is lny = y'/y. Thank you.

4. Dec 14, 2008

### Tedjn

Re: lny

It is because of the chain rule. Is that what you are asking?

5. Dec 14, 2008

### slider142

Re: lny

In case you're asking why the derivative of the natural logarithm function ln(x) is 1/x, it is because that is how the function was first defined. More precisely, the function ln(x) is defined to be the integral:
$$\int_1^x \frac{dt}{t}$$
It was defined as this integral because it was found that this integral has the properties of a logarithm, the base of which was called 'e', or Euler's number. you can read more about 'e', its discovery and its properties in this book.
The fundamental theorem of calculus then gives you the derivative of the function as 1/x, and the chain rule tells you that if you have f(x) = ln(y(x)), the derivative with respect to x is (1/y(x))*y'(x) or written implicitly, y'/y.
If you are taking the differential of the form ln(y), then you may write d(ln(y)) = dy/y.

Last edited: Dec 14, 2008
6. Dec 14, 2008

### sonofjohn

Re: lny

Thank you for the clarification.