chain rule for sqrts

[hoffmaba]

In the text I'm using (Briggs/Cochran/Gillet), both the power rule and chain rule for powers specify that n is an integer.

Despite that, you can use the power rule for sqrts, where n=1/2. (d/dx sqrt (x) = 1/2sqrt(x)).

I understand that something like sqrt(5x-2) is a composite function, but why do you have to use the chain rule for powers and not just the power rule to find f'(sqrt(5x-2))?

That is, instead of just 1/2*(5x-2)^-1/2, you have to do 1/2*(5x-2)^-1/2*5.

Are there other instances besides sqrt that I have to look out for this? Where I might be tempted to just use the power rule, but I really need the chain rule for powers? What about d/dx (5x+1)^2? Would that be 2(5x+1)*5?

Thanks!

 

[lookagain]

In the text I'm using (Briggs/Cochran/Gillet), both the power rule and chain rule for powers specify that n is an integer.

Despite that, you can use the power rule for sqrts, where n=1/2. (d/dx sqrt (x) = 1/2sqrt(x)).

I understand that something like sqrt(5x-2) is a composite function, but why do you have to use the chain rule for powers and not just the power rule to find f'(sqrt(5x-2))? See ** below.

That is, instead of just 1/2*(5x-2)^-1/2, you have to do 1/2*(5x-2)^-1/2*5.

Are there other instances besides sqrt that I have to look out for this? Yes.

Where I might be tempted to just use the power rule, but I really need the chain rule for powers? Yes.

What about d/dx (5x+1)^2? Would that be 2(5x+1)*5? Yes. And that is an answer before some further simplifying.

Thanks!

** Maybe it would help to look at this perspective:

http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-chain-2009-1.pdf


You're missing some grouping symbols. Here are some corrections:

"d/(dx) sqrt(x) = 1/[2sqrt(x)]"

". . . (1/2)*(5x - 2)^(-1/2), you have to do (1/2)*(5x - 2)^(-1/2)*5."

 

[tkhunny]

You may wish to focus on the general idea of the Chain Rule. Why would there be a separate Chain Rule for Square Roots alone?

 

[mmm4444bot]

In the text I'm using (Briggs/Cochran/Gillet), both the power rule and chain rule for powers specify that n is an integer.

Perhaps, were I to look at the text, I could understand why they wrote that.

In the Power Rule, n does not need to be an Integer; n can be any Real number. :cool:

\(\displaystyle \frac{d}{dx} \; x^{\pi} = \pi \cdot x^{\pi - 1} = \dfrac{\pi}{x} \cdot x^{\pi}\)