Question

what is derivative of ((x^6+3^x)^3)^(1/2)

Answer 1

Show us your best guess. It might be nice to simplify the exponents a little bit.

Answer 2

well im sort of confused on how to do it with sq roots but its something like \[6x^5*\ln3*3^x\]

Answer 3

That's a good guess, did you mean to put a + sign between the 6x^5 and ln3*3^x though?

Well first off, do you know how to combine the exponents? On this similar example, can you combine both exponents together?\[\Large (a^3)^{1/2}=?\]

Answer 4

if you have something with exponents raised to more exponents, let's just look at a simpler case to figure out the rule, and then we can apply what we figure out to ones where it's not so clear what to do.

\[\Large (x^3)^2=(x*x*x)^2=(x*x*x)*(x*x*x)=x^6\] try out some other examples on your own to see if you can figure it out. But it looks like we went from 3 and 2 to 6, so that seems like multiplication. Try it maybe with (x^5)^3 maybe to see if this matches.

Now apply this rule to this case. After that we'll continue on with taking the derivative.

Answer 5

Waiting on your response and I'll help you get through this and understand it if that's what you want.

Answer 6

sorry im back, i wasnt sure if u would continue before
so a^3/2

Answer 7

Alright so do you know the chain rule?
\[\Large [f(g(x))]'=f'(g(x))*g'(x)\] Can you identify what part is the inside and outside part? \[\Large(x^6+3^x)^{3/2}\]

Answer 8

so its would be z^3/2 the outside?

Answer 9

Yeah exactly, and you have already calculated the derivative of the inside. It's really about understanding this form and fitting it together correctly.

Answer 10

so \[3/2[5x^6+\ln(6)6^x ]^1/2\]

Answer 11

Almost, but what you've written is \[\Large f'(g'(x))\] but you need:\[\Large f'(g(x))*g'(x)\] So slightly different, see if you can fix it.

Answer 12

Oh also that's supposed to be ln(3) 3^x. I just wanted to make sure that's a typo so there's nothing confusing here. You're doing good. =)

Answer 13

\[3/2[5x^6+\ln(3)3^x]1/2+x^6*3^3\]

Answer 14

What is g(x)?

Answer 15

isnt that supposed to be the ln(3)*3^x

Answer 16

+x^6

Answer 17

(x^6+3^x^3)^1/2

Answer 18

Almost, except without the ln(3) part since that's part of the derivative, g'(x). You already said the outside part was the exponent, so that's good. We have all the pieces, you just need to put them together. So I'll write out a table for you of what we have and then see if you can put it together.

\[\Large f(x)=x^{3/2} \\ \Large g(x)=x^6+3^x\] What's the derivative of each of these?
\[\Large f'(x)=\frac{3}{2}x^{1/2} \\ \Large g'(x)=6x^5+\ln(3)3^x\]
\[\Large [f(g(x))]'=f'(g(x))*g'(x)\]

So basically this says the derivative is what you plug in on the right side there.

Answer 19

\[3/2[x^6+3^x]^1/2*6x^5+\in(3)3^x\]

Answer 20

Yes! I think you just have to put some parenthesis around the g'(x) part. Good job. Done! =)

Answer 21

thanks XD