Help! Differentiate and simplify the ff.

Question

anonymous · Answer

$$\log(xy)^2+\frac{ x }{ y } = \log4x^3$$

amistre64 · Answer

implicit it ....

amistre64 · Answer

each term is an individual function in its own right .... and it may be useful to know if this is partial or implicit

anonymous · Answer

i think it is implicit

amistre64 · Answer

another wrinkle is ... logs are considered in some courses as base 10,  in other courses there is only 1 real log(x) function,  the natural log.  which log does your course define 'log' as?

amistre64 · Answer

$$\large \log(xy)^2+\frac{ x }{ y } = \underbrace{log4}_{base 4?}x^3$$

anonymous · Answer

$$\log(xy)^2+\frac{ x }{ y } = \log(4x^3)$$

amistre64 · Answer

lets implicit the first term:

log[(xy)^2]

i see a chain, another chain, and a product rule ...

amistre64 · Answer

$$log(u)=\frac{u'}{u}$$
but u=a^2,  u' = 2aa'
$$log(a^2)=\frac{2aa'}{a}$$
but a = xy,  a' = x'y + xy'
$$log[(xy)^2]=\frac{2(xy)(x'y+xy')}{xy}$$

amistre64 · Answer

forgot a tiny little ^2 on that bottom a :/

amistre64 · Answer

so to correct,  the denominator will be (xy)^2 instead

amistre64 · Answer

the next term can be better written as  xy^-1  and run a product rule

and the last term is just another log run ....

anonymous · Answer

next step?