Failing to differentiate a^x • x^n by first principles

[daratmp]

Let f(x) = ax • xn

I'll omit limit because it gets unreadable quick. But it is there.

f'(x) = (f(x+h) - f(x)) / h

f'(x) = (ax+h • (x + h)n - ax • xn ) / h

we apply the binomial theorem here

f'(x) = (ax+h • (xn + hnxn-1 ....) - ax • xn) / h

And now I'm stuck. I can break down ax+h into ax • ah but that doesn't achieve much at all.

I could also express it in terms of e, but I don't know what that achieves either.

And the other thing I could do is cancel out h for hnxn-1 and the rest of the terms in the binomial expansion. But that would still leave me with h divisions.

So I still can't apply the limit.

I guess I could factorise as so:

f'(x) = (ax (ah (xn + hnxn-1 ...) - xn )) / h

But it isn't clear how to proceed, or even that proceeding is possible.

Where did I go wrong, or how can I continue?

Yes I can use the product rule but the point is to do it by first principles.

 

[stapel]

Let f(x) = ax • xn

Is there some reason that you're not simplifying [imath]ax \times xn[/imath] to [imath]anx^2[/imath]?

 

[topsquark]

Let f(x) = ax • xn

I'll omit limit because it gets unreadable quick. But it is there.

f'(x) = (f(x+h) - f(x)) / h

f'(x) = (ax+h • (x + h)n - ax • xn ) / h

we apply the binomial theorem here

f'(x) = (ax+h • (xn + hnxn-1 ....) - ax • xn) / h

And now I'm stuck. I can break down ax+h into ax • ah but that doesn't achieve much at all.

I could also express it in terms of e, but I don't know what that achieves either.

And the other thing I could do is cancel out h for hnxn-1 and the rest of the terms in the binomial expansion. But that would still leave me with h divisions.

So I still can't apply the limit.

I guess I could factorise as so:

f'(x) = (ax (ah (xn + hnxn-1 ...) - xn )) / h

But it isn't clear how to proceed, or even that proceeding is possible.

Where did I go wrong, or how can I continue?

Yes I can use the product rule but the point is to do it by first principles.

You wouldn't want to take something like this "head on," you would want to use the product rule.

-Dan

 

[Dr.Peterson]

Is there some reason that you're not simplifying [imath]ax \times xn[/imath] to [imath]anx^2[/imath]?

Apparently the problem is really about [imath]a^x x^n[/imath], not [imath]ax\cdot xn[/imath]. That doesn't simplify.

Let f(x) = ax • xn

You need to write what you mean! What you show in the title is different.

Yes I can use the product rule but the point is to do it by first principles.

That is a waste of effort. We prove theorems about the derivative in order to make it reasonable to differentiate more complicated functions using those theorems. There is no benefit in trying to do what you are asking. Were you told to do it that way?

Now, if you had to, you could use the proof of the product rule as a guide in your work. Do the same sorts of things that were done in that proof.

 

[Steven G]

Let f(x) = ax • xn

I'll omit limit because it gets unreadable quick. But it is there.

f'(x) = (f(x+h) - f(x)) / h

f'(x) = (ax+h • (x + h)n - ax • xn ) / h

Your very 1st line is wrong
Where did ax+h • (x + h)n come from? Did you mean a(x+h)*(x+h)n??
By xn do you mean xⁿ? If yes, then write x^n NOT xn as that just means x*n=n*x=nx

 

[khansaheb]

Do you know how to obtain the derivative of ln(x) - from "first principle"?