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put x in evidence

simplify it to x^-2 (1+dx/x)^-2
I hope it makes sense now

I stiil dont get it
Could someone please explain it step by step?

Ok, do you agree that x+dx could be written as x(1+dx/x)?

\[x+dx=x+\frac{x}{x} dx=x+x \cdot \frac{dx}{x}=x(1+\frac{dx}{x})\]

how can x+dx be written as x(1+dx/x)?

\[(x+dx)^{-2}=[(x)(1+\frac{dx}{x})]^{-2}=x^{-2}(1+\frac{dx}{x})^{-2} \]

Did you see what I said above what you said?

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you can factor out that x

but how did you get to x+x d/x?
the equation is (x+dx)^-2

You wanted step by step
I'm looking at the inside, x+dx, right now.

ok

If you can understand why x+dx=x(1+dx/x) we can move on to the exponent part

which i already did above

but i can redo

i dont understand why x+dx=x(1+dx/x)

where did you get the 1?

Like do you the distributive property.
It says:
a(b+c)=ab+ac

and how did you get dx/x?

x(1)=x

\[x+dx=x+\frac{x}{x} dx \]
look I multiply dx by x/x which is just one

you should see an x in both terms now

you can factor an x out

how did you get x/x?

\[x(1+\frac{1}{x}dx)\]
\[x(1+\frac{dx}{x}) \]

I just put it there so you would see better
x/x is there if you write it because it is just one

If you factor something out you are dividing the terms you factor from

im sorry but i dont understnad

*understand

For example
\[3x+9=3(\frac{3x}{3}+\frac{9}{3})=3(x+3)\]
Do you understand this example?

Can you please write down every step you do and how you get every variable?

in one answer please

If you don't understand it separately how do you expect to understand it together?

Do you understand the example I just put?

no sorry

im new to calculus

You might want to review factoring then.
Because all I did was factor 3x+9 in that example

i will

Do you know the distributive property?

no sorry

Like a(b+c) means ab+ac

yeah

That is the distributive property

ok

ok

What do you get when you multiply out the ac(b/c+1) expression?

i must go
good luck

no idea

thankyou

and dy =y'? dx =x'?

\[y+dy=(x+dx)^{-2} =x ^{-2}(1+\frac{ dx }{ x })^{-2}\]

the aim is to differentiate y=x^2

2 equal signs?

the second equal sign is the result

once we have powered the left hand side of the equation to -2

what do you mean by "ODE?"

one more question: from which course , you have this problem?

its from a book called calculus made easy, ill post the page i found it on right now

thanks for information,

no problem

ill post the page in a sec

he adds dy or dx to expand it a bit

this might clear it up
i like this book
his proofs are easier to understand than
the way was taught.