Question

I understand the idea of implicit differentiation, but actually working it out doesn't quite work out for me. Here's an example.
100x^.75y^.75= 135540
I do the product rule first and end up with 100x^.75(.25y^-.75dy/dx) + y^.25(75x^-.25)=0
i understand I have to get dy/dx by itself, but i dont see how they got 25x^.75/y^.75=-75y^.25/x^.25

Can you help?

Answer 1

25x^.75/y^.75=-75y^.25/x^.25 ?

Answer 2

i'll rewrite it

Answer 3

\[\frac{ d }{dx } 100(xy)^{\frac{ 3 }{4 }} =\frac{ 3 }{4 }100(xy)^{\frac{ -1 }{ 4 }}(y+xy \prime )\]

Answer 4

right but i dont see how they got to this part\[\frac{ 25x ^{.75} }{ y ^{.75} }* dy/dx \] = \[\frac{ -75y ^{.75} }{ x ^{.25} } \]

Answer 5

you sure you wrote the original correctly?

Answer 6

it's 100x\[100x ^{3/4}y ^{1/2} = 135,540\] it's an implicit differentiation problem

Answer 7

..

Answer 8

100x^.75y^.75= 135540

not the same...

Answer 9

o didnt relaize i wrote .75 instead of .25.

Answer 10

that's not the problem actually. I wouldn't recommend converting to decimals. you might be causing yourself confusion.

Answer 11

they gave it to me in decimal form, isnt .75=3/4

Answer 12

yes but 1/2 isn't .25

Answer 13

or .75

Answer 14

im soo gone right now lol. ur right .25=1/4

Answer 15

:)

Answer 16

100x^(3/4) y^(1/2)=135,540

so is that your problem?

Answer 17

1/4 instead of 1/2

Answer 18

ah

Answer 19

whoops dy/dx = -3y/x?

Answer 20

yes it's -3y/x. can you explain how you got there.

Answer 21

sure:

Answer 22

ignore the 100, product rule:
\[\frac{ d }{dx } ( x ^{\frac{ 3 }{4 }}y ^{\frac{ 1 }{ 4 }} = C)\]

Answer 23

ok so treat it as a constant

Answer 24

\[\huge \frac{ 3 }{4 }x ^{\frac{ -1 }{ 4 }}y ^{\frac{ 1 }{ 4 }} + x ^{\frac{ 3 }{ 4 }}\frac{ 1 }{4 }y ^{\frac{ -3 }{ 4}}y \prime =0\]

Answer 25

\[\huge x ^{\frac{ 3 }{ 4 }}\frac{ 1 }{ 4 }y ^{\frac{ -3 }{ 4 }}y \prime = -\frac{ 3 }{ 4}x ^{\frac{ -1 }{ 4 }}y ^{\frac{ 1 }{ 4 }}\]

Answer 26

ok i'm following.

Answer 27

\[\huge y \prime = \frac{ -\frac{ 3 }{ 4}x ^{\frac{ -1 }{ 4 }}y ^{\frac{ 1 }{ 4 }} }{ x ^{\frac{ 3 }{ 4 }}\frac{ 1 }{ 4 }y ^{\frac{ -3 }{ 4 }} }\]

Answer 28

make all exponents positive. divide -3/4 by 1/4

Answer 29

ok?

Answer 30

im working it out right now, hold on

Answer 31

ok i see where the 3 comes from. now i have \[\frac{ y ^{3/4} y ^{1/4} }{ x ^{3/4}x ^{1/4} }\]

Answer 32

yes

Answer 33

ie y/x

Answer 34

how do those cancel out? Is it because of the same exponent regardless of the base?

Answer 35

nevermind i got it. rules of exponent. thank you for your help

Answer 36

sure.