Chain Rule vs. Other Rules for derivatives.

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TonyMartini

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I am having a hard time understanding the difference between the chain rule and other rules for solving derivatives. Let's say I have an equation F(x) = x(x+3). Couldn't I technically define this as two different functions(F(x) = x(u) and G(u) = u+3) and do the chain rule? I know that this is incorrect but I want to know why I can't do this?

Also since the derivative of ex is simply ex why is the derivative e3x not e3x?

Any help is appreciated.
 
Personally I never use the chain rule, at least not the way it is taught using u substitution.

I use the general power rule. If you have y = ( f(x) )n, then y'= n( f(x) )n-1*f'(x)

I guess that I use the chain rule for trig function, but again I do not use this dy/dx = (dy/du)/(du/dx) stuff.
 
TonyMartini said:
I am having a hard time understanding the difference between the chain rule and other rules for solving derivatives. Let's say I have an equation F(x) = x(x+3). Couldn't I technically define this as two different functions(F(x) = x(u) and G(u) = u+3) and do the chain rule? I know that this is incorrect but I want to know why I can't do this?

Also since the derivative of ex is simply ex why is the derivative e3x not e3x?
Click to expand...

The chain rule applies to a composition of functions. The given function is not the composition of your F and G. Furthermore, x(u) is not a function of x.

The derivative of ex that you show is the derivative with respect to x. But 3x is not x.
 
As Dr. Peterson implies, the chain rule is essential when dealing with composition of functions although it applies whenever you use a u-substitution. Jomo may be right that students overuse u-substitution, but I think he is exaggerating to advise using them only in very restricted cases. I use them with complicated expressions all the time to reduce writing and mistakes, but they just add extra steps with simple expressions.

[MATH]y = x(x + 3).[/MATH]
The derivative of the above is most easily solved by the power rule.

[MATH]y = x(x + 3) = x^2 + 3x \implies \dfrac{dy}{dx}= 2x + 3.[/MATH]
A u-substitution just adds extra work without appreciable benefit.

[MATH]\text {Let } u = x + 3 \implies \dfrac{du}{dx} = 1, \ x = u - 3, \text { and }[/MATH]
[MATH]y = (u - 3)u = u^2 - 3u \implies \dfrac{dy}{du} = 2u - 3 \implies[/MATH]
[MATH]\dfrac{dy}{dx} = \dfrac{dy}{du} * \dfrac{du}{dx} = (2u - 3)(1) =[/MATH]
[MATH]2u - 3 = 2(x + 3) - 3 = 2x + 6 - 3 = 2x + 3.[/MATH]
Your second case involves a composition of functions. You must use the chain rule whether you explicitly use a u-substitution or not. But using the u-substitution explicitly ensures you will remember to use the chain rule.

[MATH]y = e^{3x}.[/MATH]
[MATH]\text {Let } u = 3x \implies \dfrac{du}{dx} = 3 \text { and } y = e^u = \dfrac{dy}{du}.[/MATH]
[MATH]\dfrac{dy}{dx} = \dfrac{dy}{du} * \dfrac{du}{dx} = e^u * 3 = 3e^u = 3 e^{3x}.[/MATH]
 
By the way, there are other applications of the chain rule besides the actual computation of derivatives of composed functions. For example when computing traveling wave partial-derivative relations, which may be derived a bit more formally by means of the chain rule
 
JeffM said:
Jomo may be right that students overuse u-substitution, but I think he is exaggerating to advise using them only in very restricted cases. I use them with complicated expressions all the time to reduce writing and mistakes, but they just add extra steps with simple expressions.
Click to expand...
Jeff, can you please give me an example where you would actually make a u-sub? I would really appreciate seeing it.
 
Jomo, I have long thought that the use of \(u\)-substitutions is greatly exaggerated as an aid for students.
I once heard a talk that conformed much of my doubts. Students learn a great deal from being forced to tease out a very complicated function.
A student who is comfortable doing a complicated derivative is equally comfortable doing a complicated antiderivative.
That means less need for \(u\)-substitutions
 
@Jomo

Here is a recent example where I actually did use substitutions. The probability of transcription error is, in my opinion, greatly reduced.

Example comparison doubt

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