Question

What's my mistake here?
GIVEN
f[x]=x/E^x^2
(f[ x +h] - f[x]) \ h

Answer 1

What's my mistake here... ?
I dont think this is the right answer

\[ \frac{(x+ h)}{e^{(x+h)^{2}}} - \frac{x}{e^{x^2}} * \frac{1}{h} \]
Find common denominator ...
\[ \frac{e^{x^2}(x+ h)}{e^{x^2}e^{(x+h)^{2}}} - \frac{x e^{(x+h)^{2}} }{e^{x^2} e^{(x+h)^{2}}} * \frac{1}{h} \]

\[ \frac{
e^{x^2} (x+ h)
- x e^{(x+h)^{2}}
}
{
e^{x^2}
e^{(x+h)^{2}}
}
* \frac{1}{h}
\]
expanded
\[ \frac{
x e^{x^2}
+ h e^{x^2}
- x e^{x^2+2hx +h^2}
}
{
e^{x^2}
e^{x^2+2hx+h^2}
}
* \frac{1}{h}
\]
exponent laws ...
\[ \frac{
x e^{x^2}
+ h e^{x^2}
- x e^{x^2}
e^{2hx}
e^{h^2}
}
{
e^{x^2}
e^{x^2}
e^{2hx}
e^{h^2}
}
* \frac{1}{h}
\]

simplify and cancel out terms
\[ \frac{
x e^{x^2}
+ h e^{x^2}
- x e^{x^2}
}
{
e^{x^2}
e^{x^2}
}
* \frac{1}{h}
\]

\[ \frac{

h e^{x^2}

}
{
e^{x^2}
e^{x^2}
}
* \frac{1}{h}
\]

\[ \frac{

h

}
{
e^{x^2}

}
* \frac{1}{h}
\]

\[ \frac{

1

}
{
e^{x^2}

}

\]

Answer 2

Your cancelling is wrong.

Answer 3

yeah, looking for another way to explain it.. without the derivative rules because this stupid course throws this problem at me in chapter 3 and doesnt introduce derivatives until chapter 4.. lol damn them

Answer 4

just thought for a minute I found a less complicated way there, by getting a common denominator. my cancelation algebra is off though I guess

Answer 5

Yeah that isn't the derivative of that

Answer 6

Down the route you went. The problem is at this step:
\[\frac{ xe^{x^{2}}+he^{x^{2}}-xe^{x^{2}}e^{2xh}e^{h^{2}} }{ e^{x^{2}}e^{x^{2}}e^{2xh}e^{h^{2}} }*\frac{ 1 }{ h } = \frac{ x + h - xe^{2xh}e^{h^{2}} }{ e^{x^{2}}e^{2xh}e^{h^{2}}}*\frac{ 1 }{ h }\]

That would be the simplification you can do, an \(e^{x^{2}}\) cancelling out.

Answer 7

....Not that helped any in terms of trying to do this with the limit definition, lol.

Answer 8

lol..

Answer 9

I got this:
\[\large \frac{1}{h}\left( {\frac{{h{e^{{x^2}}} - x{e^{{h^2}}} - x{e^{2xh}}}}{{{e^{{x^2} + {{\left( {x + h} \right)}^2}}}}}} \right)\]

Answer 10

you need chain rule to solve this

Answer 11

When I guided you through this the other day, Noted that the derivative of e^(f(x)) is f'(x)e^(f(x)) this is not really a derivative rule it is due to chain rule.

These lectures might help.
https://www.youtube.com/watch?v=8UHQhpoFF1c

The solution i gave you did not require the use of any derivative rules.

Answer 12

Here is also a detailed proof of chain rule

http://kruel.co/math/chainrule.pdf

Answer 13

true, I was reconstructing it, when I thought I had found a shortcut.. but I guess not.. because this route just cancels out to 0 again.

Answer 14

It is because it is a composite function, it is good that you are doing this though as the ability to see functions in other functions is fundamental to being able to solve derivatives

Answer 15

or rather to be able to break a function into smaller functions

Answer 16

http://tutorial.math.lamar.edu/Classes/CalcI/DerivativeProofs.aspx

Pauls notes also have an explanation of chain rule that is pretty alright

Answer 17

you can probably incorporate the chain rule proof to solve this problem

Answer 18

Not that it would help, but if you still feel the need to keep trying the limit definition route, you could try this variation and see if you can come up with some fancy simplification of it:
\[\frac{ f(x) - f(t) }{ x-t }\] as x approaches t.

Another thing to try is rewriting x as \(e^{lnx}\) and h as \(e^{lnh }\) and maybe try to write everything as exponents

Answer 19

If you prove d/dx e^(f(x)) is f'(x)e^(f(x)) then you can say your problem is completely solved.

I feel like this problem is not really something you should dwell to hard on you will likely never encounter a problem like this in real life.

Although it cant hurt to learn proofs I suppose, as it will give you a deeper understanding of the mathematics.

Answer 20

that looks like a solution actually.. prove d/dx heheh.. I have like 6 chapters of derivative related stuff coming up.. I think they just wanted me to think on this, before they give me the chain rule and power rules. but I was a bit worried if I dont get this, I wont get whats coming up. going to go watch some more vids on limits, derivatives, chain and power rules, differentials.. mind you none of those simple examples seem to deal with complex rationals and multi term exponents, so I have found myself wondering a lot, "how would I apply this?" I'm starting to see though that terms can be treated as f(x) g(x) in various ways and exponents can be broken up, so it will click soon.

Answer 21

I bet you will breeze through it, calculus is actually a pretty easy topic if you have algebra mastered.