Differentiation Product Rule

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jpanknin

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Hi all, I'd like some clarification/explanation on the product rule. I'm self-learning and using Stewart's Early Transcendentals 7E (section 3.2).

The book gives the definition of the product rule as:

[MATH]d/dx [(f(x) * g(x)] = f(x) d/dx[g(x)] + g(x) d/dx[f(x)][/MATH]
Fine so far. But the first example is:

[MATH]f(x) = xe^x[/MATH]
And you're supposed to find [MATH]f'(x)[/MATH]. Their solution is:

[MATH]f'(x) = x * d/dx (e^x) + e^x * d/dx(x)[/MATH]
I understand using the product rule if you have two functions, say [MATH]f(x)[/MATH] and [MATH]g(x)[/MATH], just like the definition, but I'm having trouble understanding why the single function [MATH]f(x)[/MATH] was split apart into [MATH]x[/MATH] and [MATH]e^x[/MATH]. Can someone please explain why these terms were separated within the same function?

Thanks in advance.
 
Thanks, Subhotosh. So to clarify, anytime there's a multiplication within a function those terms can be split out into different functions? As in:

[MATH]f(x) = x * e^x[/MATH]
Could be represented as:

[MATH]h(x) = x[/MATH] and [MATH]g(x) = e^x[/MATH]
And if [MATH]f(x) = h(x) * g(x)[/MATH] then [MATH]f(x)[/MATH] would essentially be [MATH]x * e^x[/MATH]?

And f'(x) would be:

[MATH]d/dx[(h(x)∗g(x)]=h(x)∗d/dx[g(x)]+g(x)∗d/dx[h(x)][/MATH]......................................................(1)?
 
jpanknin said:
Hi all, I'd like some clarification/explanation on the product rule. I'm self-learning and using Stewart's Early Transcendentals 7E (section 3.2).

The book gives the definition of the product rule as:

[MATH]d/dx [(f(x) * g(x)] = f(x) d/dx[g(x)] + g(x) d/dx[f(x)][/MATH]
Fine so far. But the first example is:

[MATH]f(x) = xe^x[/MATH]
And you're supposed to find [MATH]f'(x)[/MATH]. Their solution is:

[MATH]f'(x) = x * d/dx (e^x) + e^x * d/dx(x)[/MATH]
I understand using the product rule if you have two functions, say [MATH]f(x)[/MATH] and [MATH]g(x)[/MATH], just like the definition, but I'm having trouble understanding why the single function [MATH]f(x)[/MATH] was split apart into [MATH]x[/MATH] and [MATH]e^x[/MATH]. Can someone please explain why these terms were separated within the same function?

Thanks in advance.
Click to expand...
Let us change the names a little bit - to avoid confusion.

[MATH]d/dx [(h(x) * g(x)] = h(x) * d/dx[g(x)] + g(x) * d/dx[h(x)][/MATH]......................................................(1)

Notice that I have changed the name "f(x)" (in this equation ONLY) to "h(x)".

We are given:

f(x) = xex

There is an "implied" multiplication here. To be "explicitly" correct - it should be written as:

f(x) = x * ex

Now apply equation (1) with

h(x) = x .................... and

g(x) = ex

Please write back if you need further explanation.
 
jpanknin said:
Thanks, Subhotosh. So to clarify, anytime there's a multiplication within a function those terms can be split out into different functions? As in:

[MATH]f(x) = x * e^x[/MATH]
Could be represented as:

[MATH]h(x) = x[/MATH] and [MATH]g(x) = e^x[/MATH]
And if [MATH]f(x) = h(x) * g(x)[/MATH] then [MATH]f(x)[/MATH] would essentially be [MATH]x * e^x[/MATH]?

And f'(x) would be:

[MATH]d/dx[(h(x)∗g(x)]=h(x)∗d/dx[g(x)]+g(x)∗d/dx[h(x)][/MATH]......................................................(1)?
Click to expand...
Terms like - "anytime", "everytime", etc. scare me. Those are very lo_o_o_o_ng time. However, it is safe to say "most of the time" - your statement would be true.
 
If I had f(x) = 7*x2 I would NOT use the product rule even though I could.
 
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