# Thread: Trouble with Quotient Rule

1. ## Trouble with Quotient Rule

I need to simplify it further, but before I do I checked it into Mathematica and it says the derivative is incorrect. What did I do wrong?

2. ## Re: Trouble with Quotient Rule

Your calculations are correct. If you simplify you should get
1/(x^2 sqrt(1+x^2))

3. ## Re: Trouble with Quotient Rule

I find it to be a bit easier, though there is more factoring involved, if you divide by x first:
$\frac{d}{dx} \left ( - \frac{\sqrt{1 + x^2}}{x} \right ) = -\frac{d}{dx} \left ( \sqrt{ \frac{1}{x^2} + 1} \right )$

$= - \frac{d}{dx} \left ( \left [ \frac{1}{x^2} + 1 \right ] ^{1/2} \right )$

Now for the derivative:
$= -\frac{1}{2} \cdot \left ( \frac{1}{x^2} + 1 \right ) ^{-1/2} \cdot \frac{-2}{x^3}$

$= \frac{1}{x^3 \left ( \frac{1}{x^2} + 1 \right ) ^{1/2} }$

I find this result to be reasonably adequate, but your text decided differently and cleared the fraction in the denominator. So, using a little trick:
$= \frac{1}{x^2 \cdot x \left ( \frac{1}{x^2} + 1 \right ) ^{1/2} }$

$= \frac{1}{x^2 \left ( x^2 \cdot \frac{1}{x^2} + x^2 \cdot 1 \right ) ^{1/2} }$

$= \frac{1}{x^2 \left ( 1 + x^2 \right ) ^{1/2} }$

-Dan
The text passage contains a discussion about the application of the quotient rule in calculus, specifically about finding and simplifying a derivative. It includes a detailed step-by-step solution to a derivative problem.
no_question

the passage does not contain a direct mathematical question but rather a discussion and explanation of a derivative calculation.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Thread: Trouble with Quotient Rule

1. ## Trouble with Quotient Rule

I need to simplify it further, but before I do I checked it into Mathematica and it says the derivative is incorrect. What did I do wrong?

2. ## Re: Trouble with Quotient Rule

Your calculations are correct. If you simplify you should get
1/(x^2 sqrt(1+x^2))

3. ## Re: Trouble with Quotient Rule

I find it to be a bit easier, though there is more factoring involved, if you divide by x first:
$\frac{d}{dx} \left ( - \frac{\sqrt{1 + x^2}}{x} \right ) = -\frac{d}{dx} \left ( \sqrt{ \frac{1}{x^2} + 1} \right )$

$= - \frac{d}{dx} \left ( \left [ \frac{1}{x^2} + 1 \right ] ^{1/2} \right )$

Now for the derivative:
$= -\frac{1}{2} \cdot \left ( \frac{1}{x^2} + 1 \right ) ^{-1/2} \cdot \frac{-2}{x^3}$

$= \frac{1}{x^3 \left ( \frac{1}{x^2} + 1 \right ) ^{1/2} }$

I find this result to be reasonably adequate, but your text decided differently and cleared the fraction in the denominator. So, using a little trick:
$= \frac{1}{x^2 \cdot x \left ( \frac{1}{x^2} + 1 \right ) ^{1/2} }$

$= \frac{1}{x^2 \left ( x^2 \cdot \frac{1}{x^2} + x^2 \cdot 1 \right ) ^{1/2} }$

$= \frac{1}{x^2 \left ( 1 + x^2 \right ) ^{1/2} }$

-Dan
user
The text passage contains a discussion about the application of the quotient rule in calculus, specifically about finding and simplifying a derivative. It includes a detailed step-by-step solution to a derivative problem.

TYPE: NO_QUESTION

The passage does not contain a direct mathematical question but rather a discussion and explanation of a derivative calculation.