Expert: Ahmed Salami - 2/21/2006

Question
I am having trouble differentiating y=x(a to the power of x)
here's what i've done can you show me where i'm going wrong:

ln y= x ln xa = x ln a + x ln x

1/y dy/dx = ln a +(ln x + x/x)(product rule)

        = ln ax +1

dy/dx = y ln ax +y [WRONG ANSWER]

HOW DO I DIFFERENTIATE (x ln ax)

thanks  

Answer
Hi Aminah,
For y = x.a^x
lny = lnx + ln(a^x)       [logAB = logA + logB]
lny = lnx + x(lna)        [log(A^n) = n(logA)]
differentiating
1/y(dy/dx) = 1/x + lna      (lna is a constant)
dy/dx = y(1/x + lna)
     = y/x + lna
     = (x.a^x)/x + (x.a^x)lna
     = a^x + x.a^x.lna
     = a^x(1 + xlna)
I'm sure you can see where you went wrong.
I hope i have helped. You can always get back to me.
Regards.