Expert: Alon Mandes - 5/29/2009

Question
QUESTION: (1).
how do I find dy/dx of this function:
y= x^(log(6)(x))
It is log base 6 of x.

I'm not sure where to begin.


(2). This problem: 2((sin(x))^x)
I need to find f'(1).
I'm not sure how to get this also, it's like the one above. we just aren't taught enough in class on how to do different forms of the same kind of problem.

PLEASE HELP!
ANS would help with steps so I know how to figure these out.

ANSWER: (1) y= x^(log(6)(x))
We know that Logx{Base 6}=Lnx/Ln6. Thus, we get :
y=x^(Lnx/Ln6)
Let's perform Ln for both sides of the equation :
Lny=xLnx
Let's derive both sides with respect to x :
(y'/y)=(Lnx/Ln6)+x(1/x)
y'=y[(Lnx/Ln6)+]
y'=[(Lnx/Ln6)+1]x^[Lnx/Ln6] .
"Where Ln6=1.791

(2) f(x)=2(sinx)^x
Let's perform Ln :
Ln[f(x)]=xLn[2sinx]
Let's derive:
f'(x)/f(x) = Ln[2sinx] + [2xcosx]/[2sinx]
f'(x)=[2(sinx)^x]Ln[2sinx]+[2(sinx)^x][xcotx].
f'(1)=[2sin(1)]Ln[2sin(1)]+[2sin(1)][cot(1)]
f'(1)=[0.034][-3.35]+[0.034][57.28] = 1.833

Alon.



---------- FOLLOW-UP ----------

QUESTION: hi alon,
For both answers they ended up being wrong..

for 1. I put in [(Lnx/Ln6)+1]x^[Lnx/Ln6] and it didn't work and for 2. i put in 1.833 and it didn't work as well.. so i'm not sure why that is.

Answer
Hi Brooke,
I know why it didn't work, & I apologize. The solution is not
exact. Here is the right solution, for both problems :

(1) y= x^(log(6)(x))
We know that Logx{Base 6}=Lnx/Ln6. Thus, we get :
y=x^(Lnx/Ln6)
Let's perform Ln for both sides of the equation :
Lny=(Lnx/Ln6)*Lnx ("Here was the mistake")
Lny=(Ln²x/Ln6)
Let's derive both sides with respect to x :
y'/y=(2Lnx)/(xLn6)
y'=y[(2Lnx/xLn6)]
y'=[(2Lnx/xLn6)]x^[Lnx/Ln6] .
y' can also be written as :
y'=[(2Logx)/x][x^(Logx)] . ("Where the Log is to base 6")
-----------------------------------------------------

(2) f(x)=2*(sinx)^x
Let's perform Ln :
Ln[f(x)]=Ln[2*(sinx)^x]=Ln(2)+xLn[sinx] ("Here was the mistake")
Let's derive:
f'(x)/f(x) = Ln[sinx] + [xcosx]/[sinx]
f'(x)=[2(sinx)^x]Ln[sinx]+[2(sinx)^x][xcotx].
f'(1)=[2sin(1)]Ln[sin(1)]+[2sin(1)][cot(1)]

Once again, my mistake, sorry !
Alon.