Question

HELP Please. Use implicit differentiation to find the equation of the tangent line to the curve xy^3+xy=4 at the point (2,1) . The equation of this tangent line can be written in the form: y=mx+b
where M is:
and where B is:

Answer 1

remember product rule for implicit
\[(fg)' = f'g +fg'\]

Answer 2

M = slope = dy/dx

Answer 3

oh so for implicit we use the product?

Answer 4

always?

Answer 5

yes and chain rule....since it is with respect to x
\[d/dx f(y) = f'(y)*\frac{dy}{dx}\]

Answer 6

so which is f and which is g

Answer 7

im a little confused since xy^3 are together.

Answer 8

depends on the term...first there is "xy^3"
f(x) = x
g(y) = y^3

Answer 9

so f = x and g= y^3 ?

Answer 10

so is the derivative: x*3y+ y^3*1 ?

Answer 11

yes, so result is
\[1*y^{3}+ x*3y^{2} \frac{dy}{dx}\]

Answer 12

dy/dx?

Answer 13

thats what the slope is...which is what we are trying to find

Answer 14

all i know is the product rule is this:

\[\frac{ d }{ dx } (fg)= f g \prime+g f \prime\]

Answer 15

anytime you differentiate a function of "y" wrt x you must multiply by "dy/dx"
refer to my prev post

Answer 16

otay. so then what.

Answer 17

repeat process for all terms of equation

Answer 18

?

Answer 19

what do you mean

Answer 20

xy^3 +xy = 4

you have to differentiate each term, the "xy^3" and "xy" adn "4"

Answer 21

3xy^2.....

Answer 22

??
first term is
\[(y^{3} +3xy^{2} \frac{dy}{dx})\]

Answer 23

next term is "xy" .... derivative is
\[(y + x \frac{dy}{dx})\]

then of course , derivative of constant is 0
now you have
\[y^{3} +3xy^{2} \frac{dy}{dx} +y+x \frac{dy}{dx} = 0\]

Answer 24

solve for "dy/dx" in terms of x and y

Answer 25

okay i understand from here:

\[\frac{ dy }{ dx}= 3xy ^{2}+ y ^{3}\]

Answer 26

then from there, i don't know what to do.

Answer 27

@dumbcow

Answer 28

well from there you would plug in the given point (2,1) to give you the slope (M) of tangent line

but how did you solve for dy/dx .... your answer is wrong

Answer 29

i didn't i'm doing this step by step

Answer 30

from the result.. ^^ i just multiplied.

Answer 31

?? sorry i am not understanding

Answer 32

remember before you wrote "yes, result is bla blah" then from there i just multiplied... x*(3y)^2 + y^3 *1

Answer 33

you know.

Answer 34

hey mathcalculus

Answer 35

do you know chain rule