derivative.......

[hassam]

wats the derivative of lnx^sinx....?

 

[usman]

I guess u mean to ask "differentiate [ln(x)]^[sin(x)]....?" ain't it
or do u mean "differentiate ln[(x)^sin(x)]....?"
or sth else...???

 

[anillatoo]

Not sure if the power rule can be used in this case since the power is a function and not a constant...

 

[Nibz]

wats the derivative of lnx^sinx....?

By Log rule:
We have lnx^sinx = sinx.lnx
Now simple as hell
Derivative = sin(x).(1/x) + ln(x).cos(x)

 

[ahmed t]

nibz was close
after the log rule u must diffrentiate using the product rule not just cos(X).(1/x)

 

[usman]

In the above case, u all hav assumed that the question was to differentiate "ln[(x)^sin(x)]." Bt what if the question was to find the derivative of "[ln(x)]^[sin(x)]." In dat case, we could employ logarithmic differentiation of the form y = f(x)^g(x) ---> ln = ln[f(x)^g(x)] = g(x)*ln[f(x)]
Then differentiating w.r.t. x -----------> y'/y = g'(x)*ln[f(x)] + g(x)*[f'(x)/f(x)] (using product rule)
Rearranging ---------------------------> y' = y*[g'(x)*ln[f(x)] + g(x)*[f'(x)/f(x)]]
Now put f(x) = ln(x) with g(x) = sin(x) and let y = f(x)^g(x) = [ln(x)]^[sin(x)]
------> y' = y*[g'(x)*ln[f(x)] + g(x)*[f'(x)/f(x)]]
= [ln(x)]^[sin(x)]*[cos(x)*ln(ln(x)) + sin(x)/[x*ln(x)]]
= [[ln(x)]^[sin(x) - 1]/x]*[x*ln(x)*cos(x)*ln(ln(x)) + sin(x)]

 

[ahmed t]

we all assumed that way cause its in our syllabus,wheras the other way isnt.
usman wat syllabus are u studying cause in a lot of other threads questions outside the syllabus where asked an u could anser all?

 

[usman]

yar actually, m essentially studying the same syllabus as yours (i.e. CIE,& i guess most probably, dat's wt u r studying 2 ) :!: 2 b precise, m in AS level n studying Phy(9072), Chem(9701), Maths(9709) and Further Maths(9231), and will b havin AS xmx in the cming May/June session

 

[ahmed t]

so how do u know all these things that arnt in our syllabus

 

[anillatoo]

yar actually, m essentially studying the same syllabus as yours (i.e. CIE,& i guess most probably, dat's wt u r studying 2 ) :!: 2 b precise, m in AS level n studying Phy(9072), Chem(9701), Maths(9709) and Further Maths(9231), and will b havin AS xmx in the cming May/June session

That is pretty brutal the subjects you are doing! I am studying Maths, Further Maths and Physics (All CIE), won't have time to study chemistry. U doing them at A2 level as well?!

 

[usman]

yeah inshallah.....!!!