Mastering the Chain Rule: A Quick Guide for Calculus Students

[asd1249jf]

Homework Statement

y = 2x / (1+x^2)^2

Find dy/dx

Homework Equations

Chain rule

The Attempt at a Solution

I completely forgot how to apply the chain rule.. I mean, I can always apply the quotient rule, but I'm sure this is 1000 times easier if you can apply the chain rule. Do you do something like

u = 1+x^2
du = 2x

so

y = du/u

But I may be getting confused with substitution rule with integration.. it's been a while since I touched calculus.. any suggestions?

 

[arunbg]

Chain rule goes something like
dy/dx=dy/du*du/dv*...*df/dx .

It is usually used when you when you have a nested combination of functions, ie functions within functions.

For your question, you need to use both the quotient rule as well as the chain rule ( (1+x^2)^2, which is the funtion 1+x^2 within a squaring function ) .

Can you finish your problem now ?

 

[asd1249jf]

Chain rule goes something like
dy/dx=dy/du*du/dv*...*df/dx .

It is usually used when you when you have a nested combination of functions, ie functions within functions.

For your question, you need to use both the quotient rule as well as the chain rule ( (1+x^2)^2, which is the funtion 1+x^2 within a squaring function ) .

Can you finish your problem now ?

Right.. that's the BRUTE force way to do it.

I was wondering if there was a way to finish this without even applying the quotient rule

 

[suspenc3]

if you don't want to use the quotient rule you can bring the denominator up top so:$$y= 2x (1+x^2)^{-2}$$ now use the product rule

 

[cristo]

Right.. that's the BRUTE force way to do it.

I was wondering if there was a way to finish this without even applying the quotient rule

Yes, just write the expression as 2x(1+x²)^-2, and use the product rule on this.

 

[asd1249jf]

if you don't want to use the quotient rule you can bring the denominator up top so:$$y= 2x (1+x^2)^{-2}$$ now use the product rule

I guess that's true, but when I saw the relating terms, I was thinking this could be solved by ONLY using chain rule.

I guess it's impossible.