Why did they change the sign to be negative?

In summary: Thanks for catching that.In summary, the conversation discusses differentiating the function y=1/x using the power rule. The bottom line of the image incorrectly shows the outcome as x-1/x2 instead of 1-1/x2. The conversation also points out that the image is poorly worded and can be confusing. The correct outcome of the differentiation is -1/x^2.
  • #1

JR Sauerland

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http://imgur.com/BNQPAQa

I'm missing something. Why does x+1/x suddenly become x-1/x2? I get that they moved the x to the top, subtracted the -1-1, but does that exclusively set the fraction negative all on its own ? I've computed it by hand. 1-2 comes out to be 1... So why is it negative... ?
 
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  • #2
Starting with ##y=1/x## then you can write it as ##y=x^{-1}## and then using the rule ##dy/dx = d/dx(x^n) = nx^{n-1}## you get ##dy/dx=(-1)x^{-1-1} = -x^{-2} = -1/x^2##

In the bottom line of the picture, they are evaluating ##d/dx(x + 1/x)## as an example which is ##d/dx(x) + d/dx(1/x) = 1 -1/x^2##
 
  • #3
JR Sauerland said:
http://imgur.com/BNQPAQa

I'm missing something. Why does x+1/x suddenly become x-1/x2?
You're missing two things:
1) It isn't "x - 1/x2" It's ##1 - \frac 1 {x^2}##. At the very least,use ^ to indicate an exponent, as in 1 - 1/x^2.
2) They are differentiating x + 1/x to get It's ##1 - \frac 1 {x^2}##, but they wrote what they're doing very poorly, IMO.
##\frac d {dx}(x + \frac 1 x) = \frac d {dx} x + \frac d {dx} x^{-1} = 1 + (-1)x^{-2} = 1 - \frac 1 {x^2}##

JR Sauerland said:
I get that they moved the x to the top, subtracted the -1-1, but does that exclusively set the fraction negative all on its own ? I've computed it by hand. 1-2 comes out to be 1... So why is it negative... ?
 
  • #4
The image is extremely awkwardly worded. You should not feel bad for not getting it. It is just a terrible way of teaching this topic.

What happened is that it recognized that ##1/x## can be re-written as x^-1, and then you apply the power rule of differentiation: derivative of ##x^n## is nx^(n-1).

In our case, n = -1, so nx^(n-1) is -1 * x ^ (-1 -1) = - x^-2 = 1/x^2
 
  • #5
FQVBSina said:
The image is extremely awkwardly worded. You should not feel bad for not getting it. It is just a terrible way of teaching this topic.

What happened is that it recognized that ##1/x## can be re-written as x^-1, and then you apply the power rule of differentiation: derivative of ##x^n## is nx^(n-1).

In our case, n = -1, so nx^(n-1) is -1 * x ^ (-1 -1) = - x^-2 = 1/x^2
At the end you lost the sign. It should be ##\frac {-1} {x^2}##
 
  • #6
Mark44 said:
At the end you lost the sign. It should be ##\frac {-1} {x^2}##
Yes I did. The equation maker here is very bad with exponents, got me all messed up.
 

1. Why is the sign now negative instead of positive?

The sign was likely changed to negative because of a change in the direction or orientation of the variable being measured. In mathematics and science, negative numbers are used to represent values that are less than zero, while positive numbers are used to represent values that are greater than zero.

2. How does changing the sign affect the meaning of the data?

Changing the sign of a variable can affect the interpretation of the data. For example, a positive sign may indicate an increase or addition, while a negative sign may indicate a decrease or subtraction. It is important to carefully consider the context and units of the data when interpreting the meaning of a changed sign.

3. Can changing the sign impact the results of the study?

Yes, changing the sign can impact the results of a study. Depending on the specific research question and methodology, changing the sign of a variable can alter the conclusions and implications of the study. It is important for scientists to carefully track and document changes in variables to ensure the accuracy and validity of their results.

4. Is changing the sign a common occurrence in scientific research?

Yes, changing the sign of a variable is a common occurrence in scientific research. This is because variables can be measured in different directions or orientations, and it is important for scientists to accurately represent these changes in their data. Additionally, changing the sign can reveal important patterns or relationships that may not have been apparent with the original sign.

5. How do scientists decide when to change the sign of a variable?

Scientists make the decision to change the sign of a variable based on the specific research question and methodology being used. In some cases, the direction of the variable may be clear and straightforward, while in others it may require careful consideration and analysis. Ultimately, the decision to change the sign should be based on maximizing the accuracy and validity of the research results.

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