# Complicated derivative question

• sooyong94
In summary, to find the derivative of ##y=\sqrt{\sqrt{(1+\sqrt{x})}-1}##, you can take the natural log of both sides and use the chain rule to find the derivative in terms of x. Be careful when expanding the natural log.
sooyong94

## Homework Statement

How to I find the derivative of
##y=\sqrt{\sqrt{(1+\sqrt{x})}-1}##

## Homework Equations

Logarithmic derivative

## The Attempt at a Solution

I decided to take natural logs to both sides, and this is what I have:
##\ln y=\frac{1}{2} \ln (\sqrt{(1+\sqrt{x})}-\frac{1}{2}##
##\ln y=\frac{1}{4} \ln(1+\sqrt{x})-\frac{1}{2}##
Then an implicit derivative gave me this:
##\frac{1}{y} \frac{dy}{dx}=\frac{1}{8} (\frac{1}{\sqrt{x}+x})##

Now when I multiply both sides by y, the expression becomes very complicated. How do I manipulate them? :(

Why would you want to simplify the expression?
Just leave it as it is.

The reason is I need to find dy/dx in terms of x. :(

Then substitute the expression for y, and that's it.

Chain rule seems complicated... :(

sooyong94 said:
I decided to take natural logs to both sides, and this is what I have:
##\ln y=\frac{1}{2} \ln (\sqrt{(1+\sqrt{x})}-\frac{1}{2}##
You made a mistake there.
##\ln y=\frac{1}{2} \ln ( \sqrt{(1+\sqrt{x})}-1)##

Wasn't it looks the same when expanded?

sooyong94 said:
Wasn't it looks the same when expanded?
When expanding what? ##\ln(a+b) \neq \ln(a) + \ln(b)##

Oops, my bad. :P

## 1. What is a complicated derivative?

A complicated derivative is a mathematical concept that involves calculating the rate of change of one variable with respect to another variable. It is used to describe how a function changes as its input changes.

## 2. How do I solve a complicated derivative?

To solve a complicated derivative, you need to use a combination of mathematical rules and techniques, such as the power rule, product rule, quotient rule, and chain rule. It is also helpful to have a deep understanding of algebra and calculus.

## 3. Why are complicated derivatives important?

Complicated derivatives are important because they have a wide range of applications in physics, engineering, economics, and other fields. They are used to model and analyze real-world situations and make predictions about how systems will behave.

## 4. What are some common mistakes when solving complicated derivatives?

Some common mistakes when solving complicated derivatives include forgetting to apply the chain rule, using incorrect signs, and making algebraic errors. It is important to carefully follow the rules and double-check your work to avoid these mistakes.

## 5. Is there a shortcut to solving complicated derivatives?

While there is no universal shortcut for solving complicated derivatives, there are some tips and tricks that can make the process easier. These include simplifying the function before taking the derivative, using the product and quotient rules sparingly, and being familiar with common derivatives of basic functions.

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