# Derivatives of ln(x): Taking on Complex Problems

• jumbogala
In summary, the conversation discusses how to take the derivative of ln(x) and how to apply it to more complicated expressions, such as ln(x4(2x+5)5) and ln[x5(x+4)3(x3+4)6]. The process involves using the power rule and the chain rule to find the derivative of each term before combining them. It is also noted that any term can be chosen as u, v, or w, but the natural log function must not be forgotten when taking the derivative.
jumbogala

## Homework Statement

I know how to take the derivative of ln(x), it's just 1/x. But what if you had something more complicated than just x?

For example, ln(x4(2x+5)5)?

## The Attempt at a Solution

I guess you would still do 1/(x4(2x+5)5), then multiply it by the derivative of the denominator.

Which would be 4x3(2x+5)5 + x4(5(2x+5)4)(2).

Is that correct?

The problem I'm supposed to be doing is actually more complicated, it's ln[x5(x+4)3(x3+4)6]. Would the procedure be similar? I guess I'm not sure about taking the derivative of something with 3 terms, I've only ever seen it done with two.

Yes. You are correct:

For example:

y= ln(x2)

y' = (1 / x2) .dx2dx

y' = 2x/x2

Also with three terms:

y= u.v.w

y' = u'.v.w + u.v'.w + u.v.w'

or you can deal with u and v and then w.
I think the u'.v.w + u.v'.w + u.v.w' will be a bit quicker

let u = x5 , v= (x+4)3 , w = (x3+4)6

(Note that any of these could be u, v w) But remember not to forget the natural log function when differentiating :)

Last edited:

## 1. What is the derivative of ln(x)?

The derivative of ln(x) is 1/x. This can be derived using the chain rule and the fact that the derivative of ln(x) is equal to 1/x.

## 2. How do you take the derivative of ln(x) when x is a complex number?

The derivative of ln(x) when x is a complex number can be found using the Cauchy-Riemann equations, which relate the partial derivatives of the real and imaginary parts of a complex function. Specifically, the derivative of ln(x) is given by 1/x + i*0, where i is the imaginary unit.

## 3. What are the applications of derivatives of ln(x)?

Derivatives of ln(x) have many applications in mathematics, physics, and engineering. They are commonly used in optimization problems, such as finding the maximum or minimum value of a function. They are also used in calculating rates of change and in solving differential equations.

## 4. Can the derivative of ln(x) be negative?

Yes, the derivative of ln(x) can be negative. This occurs when x is a negative real number or a complex number with a negative real part. The derivative of ln(x) is always negative for values of x between 0 and 1, and it is positive for values of x greater than 1.

## 5. Is it possible to take the second derivative of ln(x)?

Yes, it is possible to take the second derivative of ln(x). The second derivative of ln(x) is equal to -1/x^2. This can be found using the quotient rule and the fact that the derivative of ln(x) is 1/x.

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