Derivative and Integral Question

[Oneiromancy]

1. y = 3^^{}log_{}_{}_2^{}(t) Find dy/dx ... I tried using logarithmic differentiation but that didn't work.

2. \int x^{2x}(1 + ln x)dx\ ... I set u = x^{2x} but my du didn't quite work out right.

 

[Dick]

a^b=e^(b*ln(a)). log_a(b)=ln(b)/ln(a). I don't think you are trying hard enough. Try again, and show us what you tried this time if you are still having problems.

 

[quantumdude]

1. y = 3^^{}log_{}_{}_2^{}(t) Find dy/dx

Are you sure? Unless t is some function of x, that derivative is identically zero.

 

[Oneiromancy]

I meant dy/dt. :)

I will try to solve these again a little later, thanks.

 

[lurflurf]

use the power rule
(u^v)'=v*u^(v-1)*u'+u^v*log(u)*v'
or
[log(u^v)]'=v{[log(u)]'+log(u)[log(v)]'}

 

[Dick]

use the power rule
(u^v)'=v*u^(v-1)*u'+u^v*log(u)*v'
or
[log(u^v)]'=v{[log(u)]'+log(u)[log(v)]'}

I wouldn't recommend you do that unless you are willing to carry those formulas around in your head for the rest of your life. You don't need them. Sorry, lurflurf. Did you really remember those, or did you derive them just now??

 

[Oneiromancy]

I understand how to get the answer now, thanks.

 

[lurflurf]

I wouldn't recommend you do that unless you are willing to carry those formulas around in your head for the rest of your life. You don't need them. Sorry, lurflurf. Did you really remember those, or did you derive them just now??

I really remember those.
I think if you are going to remember 15-20 differentiation formulas that would be one of them.
If one recalls
(u^v)'=v*u^(v-1)*u' when v'=0 obvious since [x^a]'=a*x^(a-1)
(u^v)'=u^v*log(u)*v' when u'=0 obvious since [exp(a*x)]'=a*exp(a*x)
one knows by the chainn rule the sum generalizes to the case where neither is constant
it is actually easier to recall
(u^v)'=v*u^(v-1)*u'+u^v*log(u)*v'
that two separate equations

One must decide how many calculus formulas to carry in ones head (or written on the back of ones hand) at different points in life.
That formula is more helful than say
[log(sin(a*x))]'=a*cot(a*x)
and less useful than
c'=0
your results may very
in particular if I ever forget that formula it is easer to derive the general result when needed and apply it than to derive a special case.

 

[Dick]

Interesting. I just do (u^v)'=[e^(ln(u)*v)]' and go from there. But you are quite right. The number and selection of calculus formulas you need to carry in your head is a personal choice. But I don't think I'd ever even seen those before.

 

[DavidWhitbeck]

1. y = 3^^{}log_{}_{}_2^{}(t) Find dy/dx ... I tried using logarithmic differentiation but that didn't work.

Just simplify your expression before taking the derivative.

2. \int x^{2x}(1 + ln x)dx\ ... I set u = x^{2x} but my du didn't quite work out right.

Have you ever found the derivative of x^x before? Knowing that could be quite helpful.