# Should be simple differentation

• May 15th 2011, 06:55 AM
Should be simple differentation
But I cant do it :(

$\displaystyle F(x) = x{a}^{ 1-x}$ Differentiate with respect to x, so $\displaystyle u = x$ and $\displaystyle v = {a}^{ 1-x}$ But i Cant Differentiate v :(
• May 15th 2011, 06:57 AM
TheEmptySet
Quote:

Originally Posted by adam_leeds
But I cant do it :(

$\displaystyle F(x) = x{a}^{ 1-x}$ Differentiate with respect to x, so $\displaystyle u = x$ and $\displaystyle v = {a}^{ 1-x}$ But i Cant Differentiate v :(

use algebra

$\displaystyle v=a\cdot a^{-x}=ae^{\ln(x^{-1})}= ae^{-\ln(x)}$

Can you finish from here?
• May 20th 2011, 04:05 AM
Quote:

Originally Posted by TheEmptySet
use algebra

$\displaystyle v=a\cdot a^{-x}=ae^{\ln(x^{-1})}= ae^{-\ln(x)}$

Can you finish from here?

$\displaystyle \frac{-a}{ x} {e}^{ -ln(x)}$

?
• May 20th 2011, 04:57 AM
Ackbeet
Quote:

Originally Posted by TheEmptySet
use algebra

$\displaystyle v=a\cdot a^{-x}=ae^{\ln(x^{-1})}= ae^{-\ln(x)}$

Can you finish from here?

I think there is an algebra error here. I get

$\displaystyle v=a\cdot a^{-x}=a\,e^{\ln(a^{-x})}=a\,e^{-x\ln(a)}}.$
• May 20th 2011, 08:09 AM
zebode
Quote:

Originally Posted by Ackbeet
I think there is an algebra error here. I get

$\displaystyle v=a\cdot a^{-x}=a\,e^{\ln(a^{-x})}=a\,e^{-x\ln(a)}}.$

I think too.. I get this same thing...
• May 20th 2011, 11:18 AM
Quote:

Originally Posted by Ackbeet
I think there is an algebra error here. I get

$\displaystyle v=a\cdot a^{-x}=a\,e^{\ln(a^{-x})}=a\,e^{-x\ln(a)}}.$

How do you differentiate that wrt to x?
• May 20th 2011, 11:22 AM
Ackbeet
Well, the same way you'd differentiate exp(ax): by using the chain rule. The expression -ln(a) is just a constant, right?
• May 21st 2011, 06:11 AM
Quote:

Originally Posted by Ackbeet
Well, the same way you'd differentiate exp(ax): by using the chain rule. The expression -ln(a) is just a constant, right?

Yeah

so it is $\displaystyle -aln(a){e}^{-xln(a) }$
• May 21st 2011, 06:15 AM
The answer for dv should be $\displaystyle -{a}^{1-x } lna$
$\displaystyle -a \ln(a)e^{-x\ln(a)}=-a \ln(a)e^{\ln(a^{-x})}= -a\ln(a) \cdot a^{-x}=-a^{1-x}\ln(a)$