Question

1. Why can’t you use the Power Rule to compute f'(x) for f(x) = (x+1)^x ? I know that we can't, because it's a restriction of the rules, but how can I phrase this in words?

2. Why can’t you use a formula like d/dx 3^x = 3^x ln 3 to compute f'(x)?

I am very familiar with derivatives. However, for these two questions, I am unable to phrase the answers.

Answer 1

Perhaps touching on the idea of Power Rule and of the Exponential derivative;
- Power Rule deals with functions whose base varies with respect to x and exponent is a constant.
- Exponential derivative deals with functions whose base is a constant and the exponent is changing with respect to x.
The function we have is changing in both places, yet each rule would just yield a different answer.

I think there is a better way to phrase this yet, its just an initial reaction. I am considering if limit definition makes it more clear yet...

Answer 2

^^^ concur with the above answer.
This is a case of finding the derivative of a function that has a variable being raised to a variable.
The power rule applies to finding the derivative a function that has a variable being raised to a constant.
In the second case, 3^x is a constant being raised to a variable and cannot be applied here because we have a variable being raised to a variable.

Answer 3

I don't understand
2. Why can’t you use a formula like d/dx 3^x = 3^x ln 3 to compute f'(x)?

\[ 3 = e^{\ln(3)} \\ 3^x= \left( e^{\ln(3)} \right)^x = e^{x\ln(3)}
\]
and you can take the derivative of e^(ax)

Answer 4

and that idea also works for (x+1)^x

Answer 5

@AccessDenied: Interesting insight.... but yes, there's probably an even more definitive answer.

@phi: Yes, that's what I thought. I was second-guessing myself, and then I was thinking that it could be a trick question....

Answer 6

\[ (x+1)^x = e^{x \ln(x+1)} \]
which you can differentiate to get
\[ e^{x \ln(x+1)} \frac{d}{dx}\left( x \ln(x+1)\right) = (x+1)^x \left(\ln(x+1)+\frac{x}{x+1} \right)\]

Answer 7

I think what the question is asking is:
If f(x) = (x+1)^x
Why isn't f'(x) = x * (x+1)^(x-1)? (much like the derivative of (x+1)^4 = 4(x+1)^3)

And if f(x) = (x+1)^x
Why isn't f'(x) = (x+1)^x * ln(x+1)? (much like d/dx 3^x = 3^x ln 3)

Answer 8

That process works from what I can see.... I think if we have this rule given: \( \dfrac{d}{dx} a^x = a^x \ln a \),it doesn't cover the chain rule part. Then since we need this auxiliary rule to finish the derivative, the formula does not QUITE work out.

Answer 9

so we have to interpret the question (to part 2) as,
explain why if we write the expression as a power to e, and differentiate it incorrectly (i.e. neglect the chain rule) then it won't work....

Answer 10

I am sticking to my original answer for this question:
1. We cannot apply the power rule here because the exponent is not a constant.
2. We cannot apply the derivative of 3^x rule here because the base is not a constant.

Answer 11

In most situations I see that formula in particular, a is just assumed constant. The original answers are true then; although if you understand phi 's process is generally true, this question is hardly important in the long run. Just know not to rely on the two formulas without understanding them. :)

Answer 12

Thanks everyone!