# How to differentiate y = ln( 1+x^2)^1/2

• geffman1
In summary, the question asks for help differentiating y=ln(1+x^2)^1/2. The correct answer is x/(1+x^2). The solution involves using the power rule and chain rule. It is important to note that the function must be written as ln sqrt(1+x^2) in order to get the correct answer. The 1/2 does not disappear when differentiating.

## Homework Statement

hey guys, got a question. how do you differentiate y=ln(1+x^2)^1/2. any help would be appreciated, thanks

## The Attempt at a Solution

Power rule and chain rule.

To get the answer you have listed your function must be $\ln \sqrt{1+x^2}$ and not $\sqrt{\ln(1+x^2)}$. You can write $\ln \sqrt{1+x^2}=\frac{1}{2}\ln (1+x^2)$, perhaps this form is less intimidating?

so the 1/2 just stays out the front without it being differentiate, i thought it dissapeared? thanks for the replys

geffman1 said:
so the 1/2 just stays out the front without it being differentiate, i thought it dissapeared? thanks for the replys
Go check your rules of differentiation again...

## 1. What is the derivative of y = ln(1+x^2)^1/2?

The derivative of y = ln(1+x^2)^1/2 is given by the chain rule:
dy/dx = (1/2)(1+x^2)^(-1/2)(2x)(1+0) = x/(1+x^2)^1/2

## 2. How do you simplify the expression ln(1+x^2)^1/2?

To simplify the expression, you can use the power rule for logarithms:
ln(a^b) = b ln(a).
Applying this rule, we get ln(1+x^2)^1/2 = (1/2)ln(1+x^2).

## 3. What is the domain of y = ln(1+x^2)^1/2?

The domain of y = ln(1+x^2)^1/2 is all real numbers, since both the natural logarithm and the square root function are defined for all positive real numbers.

## 4. Can the expression ln(1+x^2)^1/2 be rewritten in a different form?

Yes, ln(1+x^2)^1/2 can also be written as ln√(1+x^2) or ln(√(1+x^2)). Both forms are equivalent.

## 5. How can the derivative of y = ln(1+x^2)^1/2 be used in real-life applications?

The derivative of y = ln(1+x^2)^1/2 can be used to find the rate of change of a function that involves the natural logarithm and the square root. This can be useful in fields such as physics, economics, and engineering where logarithmic and exponential functions are commonly used to model real-life phenomena.