# Derivative and Integral Question

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.

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Dick
Homework Helper
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.

Tom Mattson
Staff Emeritus
Gold Member
1. y = 3^$$^{}log_{}_{}_2^{}(t)$$ Find dy/dx

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

I meant dy/dt. :)

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

lurflurf
Homework Helper
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)]'}

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Dick
Homework Helper
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??

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I understand how to get the answer now, thanks.

lurflurf
Homework Helper
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 seperate 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
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
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}$$ but my du didn't quite work out right.
Have you ever found the derivative of $$x^x$$ before? Knowing that could be quite helpful.