Need to differentiate y = x^(log base5 (x)) in terms of natural log; already tried

So, I've got such a function: y = x^(log base5 (x)), which I have to differentiate in terms of ln;

I started by solving the equation y = x^(lnx/ln5) = x^(lnx - ln5) (wonder if here is my mistake)

after that ln y = ln x^(lnx - ln5)=> 1/y*y' = (lnx - ln5)'(ln x) + (ln x)'(lnx - ln5) => (1/x - 1/5)(ln x) + (1/x)(lnx - ln5) =>

y' = x^(lnx - ln5)[(1/x - 1/5)(ln x) + ((lnx - ln5)/x)], that's what I got, but putting it in webwork it consideres it as incorrect,

tell me please were is my mistake,

Thanks,

Re: Need to differentiate y = x^(log base5 (x)) in terms of natural log; already tri

Quote:

Originally Posted by

**dokrbb** So, I've got such a function: y = x^(log base5 (x)), which I have to differentiate in terms of ln;

I started by solving the equation

**y = x^(lnx/ln5) = x^(lnx - ln5) (wonder if here is my mistake)** Yea this is your mistake. you would only be able to do this if it was
after that ln y = ln x^(lnx - ln5)=> 1/y*y' = (lnx - ln5)'(ln x) + (ln x)'(lnx - ln5) => (1/x - 1/5)(ln x) + (1/x)(lnx - ln5) =>

y' = x^(lnx - ln5)[(1/x - 1/5)(ln x) + ((lnx - ln5)/x)], that's what I got, but putting it in webwork it consideres it as incorrect,

tell me please were is my mistake,

Thanks,

Carrying on from

Now just do some implicit differentiation and use the product rule on the right hand side to get your answer.

Re: Need to differentiate y = x^(log base5 (x)) in terms of natural log; already tri

so, I assume the right answer would be

y'/y = [((1/x)ln 5 - ln x (1/5))/(ln5)^(2)]*ln x + (lnx/ln5)*(1/x) =>

y' = (lnx/ln5)*ln(x) {[((1/x)ln 5 - ln x (1/5))/(ln5)^(2)]*ln x + (lnx/ln5)*(1/x)}

y' = (lnx/ln5)*ln(x){[((1/x)ln 5 - ln x (1/5))/(ln5)^(2)]*ln x + ((x lnx/ln5)}, there is a way to contract it a little bit or its better to leave it like this?

Re: Need to differentiate y = x^(log base5 (x)) in terms of natural log; already tri

oh, let me just continue, y' = (lnx/ln5)*ln(x){[((1/x)ln 5 - ln x (1/5))/(ln5)^(2)]*ln x + ((x lnx/ln5)}, =>

y' = (lnx/ln5)*ln(x){[ (ln 5/x) - (ln x /5)/(ln5)^(2)]*ln x + ((x lnx/ln5)}, =>

y' = (lnx/ln5)*ln(x) [((lnx/((ln5)^(2))*(ln5/x)-(lnx/5)) + ((x lnx/ln5)]

does that makes sense?

Re: Need to differentiate y = x^(log base5 (x)) in terms of natural log; already tri

It's hard to read your equations, so I'll try to translate them:

Here's how I would have done it:

- Hollywood