Question

Find the slope of the tangent line to the curve -4x^2-4xy-3y^3 at the point (6,2)

Answer 1

same process as the last problem... how far did you get?

Answer 2

so we find the derivitive of 4x^2-4xy-3y^3?

Answer 3

you use implicit differentiation, yes... to find y'

Answer 4

how would i set it up?

Answer 5

not sure what you're asking... did you use implicit differentiation? what result did you get?

Answer 6

i didn't, im confused on how to use implicit differentiation

Answer 7

you treat y as some function of x. we don't really need to know what exactly that function of x actually is... we don't care in fact. It seems weird but it works. Go through term by term and differentiate with respect to x...
eg.: first term

(-4x^2 ) '

pretty easy:
-8x

second term (-4xy) '
now we have a product of x and y
use the product rule:

-4 ( x'y +y'x)

x' is just 1
y' is y' (we don't know explicitly what function of x y is... doesn't matter, we just leave it as y')
so
(-4xy)' = -4(y +y'x)

Answer 8

Isn't there a missing term in the equation of the curve?

Answer 9

third term... use the chain rule (since y is a function of x)

try it: ( -3y^3) ' =...?

Answer 10

-9y^2?

Answer 11

pretty good, but y' "pops out", because of the chain rule...
so -9y^2 * y'

Answer 12

ohh,

Answer 13

anyway... if your original equation was something like -4x^2-4xy-3y^3 =C
we now have the result of differentiating that equation with respect to x:
-8x -4(y +y'x) -9y^2 * y' = 0

Answer 14

and now we solve for y prime?

Answer 15

all that's left to do is solve for y', which is just a matter of algebra :)

Answer 16

yep:)

Answer 17

im getting -14/3 but its coming out incorrect

Answer 18

you can imagine y is some well known function of x and take the derivative and you'll see that it works...

eg.: if y = sinx

(-3(sin(x))^3 )' = -9(sin(x))^2 * cos(x) { -9 y ^2 * y' ) !

(I left out a 'prime' in the original, fixt)

Answer 19

what did you get as the answer what i got s showing as incorrect

Answer 20

nevermind, i found what i did wrong. Thanks for the help!

Answer 21

cool.:)