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#### DeusAbscondus

##### Active member

- Jun 30, 2012

- 176

Hi folks,

I don't know if my experience is at all common (and I would like some feedback on this if possible), but I can't seem to nail down the properties of euler's number in the context of chain rule problems.

Here is the nub of my difficulty:

1. $\text{If }f(x)=e^x \text{then }f'(x)=e^x$

This I

But the following I do

2. $\text{If }g(x)=e^{x^{4}} \text{then }g'(x)=4e^{x^4}x^3$

Am I on the right track to observe that, if 2. is correct, then g(x) is a composite function, hence subject to the chain rule?

And if so, is the following a generally valid way to work through this and all such problems:

$\text{If }g(x)=e^{x^{4}} \text{find }g'(x)$

$\text{Now}g'(x)=u'v' \text{by Chain Rule}$

$\text{So, let }u=x^4 \text{and }v=e^u$

$\text{Then }u'=4x^3 \text{and }v'=e^u \text{ (by some rule which currently exceeds my understanding)}$

$\text{Therefore }g'(x)=u'v'=4x^3*e^u=\text{(via substitution) }4x^3*e^{x^{4}}$

$\text{Which, simplified }=4e^{x^{4}}x^3$

Finally, I have a similar hesitation/scruple/sense of vertigo when it comes to dealing with another unusual derivative, that of:

$ln(x)$

If anyone can see why, having the read the foregoing, I would feel unsure of myself around this animal, could they possibly add some notes to help me tame it?

Regs,

DeusAbs

I don't know if my experience is at all common (and I would like some feedback on this if possible), but I can't seem to nail down the properties of euler's number in the context of chain rule problems.

Here is the nub of my difficulty:

1. $\text{If }f(x)=e^x \text{then }f'(x)=e^x$

This I

*accept*, though, not having seen a formal proof of it, and since it is counter-intuitive, I must take it on faith.But the following I do

**not**understand; could someone help me towards understanding?2. $\text{If }g(x)=e^{x^{4}} \text{then }g'(x)=4e^{x^4}x^3$

Am I on the right track to observe that, if 2. is correct, then g(x) is a composite function, hence subject to the chain rule?

And if so, is the following a generally valid way to work through this and all such problems:

$\text{If }g(x)=e^{x^{4}} \text{find }g'(x)$

$\text{Now}g'(x)=u'v' \text{by Chain Rule}$

$\text{So, let }u=x^4 \text{and }v=e^u$

$\text{Then }u'=4x^3 \text{and }v'=e^u \text{ (by some rule which currently exceeds my understanding)}$

$\text{Therefore }g'(x)=u'v'=4x^3*e^u=\text{(via substitution) }4x^3*e^{x^{4}}$

$\text{Which, simplified }=4e^{x^{4}}x^3$

Finally, I have a similar hesitation/scruple/sense of vertigo when it comes to dealing with another unusual derivative, that of:

$ln(x)$

If anyone can see why, having the read the foregoing, I would feel unsure of myself around this animal, could they possibly add some notes to help me tame it?

Regs,

DeusAbs

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