How to differentiate 6^(sin(pix))

[Asphyxiated]

Homework Statement

Find the derivative to:

y = 6^{sin(\pi x)}

Homework Equations

You tell me

The Attempt at a Solution

I am quite surprised that we are asked to do this when we are just learning the chain rule. We have not learned how to differentiate something like a^x yet and somehow I am suppose to do this with the chain rule... If you could either let me know what the general rule is here or point me to some place that will tell me what it would be. I am not even sure what you would call such a rule, the base rule perhaps? ha

thanks!

 

[Dick]

6^(sin(pi*x))=e^(log(6)*sin(pi*x)), right? That's really not that much harder than differentiating a^x=e^(log(a)*x). And it does use the chain rule.

 

[Asphyxiated]

Ah I was unaware that

6^{sin(\pi x)} = e^{log(6)sin(\pi x)}

I found an example problem though in the book after posting this and I found the derivative to be:

y' = \pi 6^{sin(\pi x)}(ln(6))(cos(\pi x))

which does not appear to be what you suggested doing, as the derivative does not contain an e term at all.

I just follow the book step by step so I am unaware of how the natural log of 6 actually comes in here. Could you possibly clarify what exactly went on here with my solution?

thanks Dick!

 

[Dick]

Actually, it's better to do a^x first. f(x)=a^x=e^(log(a)*x). f'(x)=e^(log(a)*x)*(log(a)*x)' (chain rule, right?) which is e^(log(a)*x)*log(a). Which is (a^x)*log(a), right? Notice that the 'e' doesn't appear in the initial problem nor is it in the answer. But it appears in the middle. Same with your problem.

 

[Dick]

Ah I was unaware that

6^{sin(\pi x)} = e^{log(6)sin(\pi x)}

I found an example problem though in the book after posting this and I found the derivative to be:

y' = \pi 6^{sin(\pi x)}(ln(6))(cos(\pi x))

which does not appear to be what you suggested doing, as the derivative does not contain an e term at all.

I just follow the book step by step so I am unaware of how the natural log of 6 actually comes in here. Could you possibly clarify what exactly went on here with my solution?

thanks Dick!

And the log(6) comes in because 6=e^(log(6)). So 6^x=e^(log(6)*x).

 

[Asphyxiated]

Thanks! Very clear, I got it now.