Question

Use implicit differentiation to find the equation of the tangent line to the curve at

x^2 + xy + y^2 = 3 at (1,1)

How does this problem start? i know we need to find the slope and b, etc right? but not sure how to work the problem... please explain! Thanks!

Answer 1

The first step is use implicit differentiation to find dy/dx

can you do that ?

Answer 2

umm, do I do this?

dy
--- = chain rule?
dx

Answer 3

\[ \frac{d}{dx}\left( x^2 + xy + y^2 = 3\right) \\
\frac{d}{dx}x^2 + \frac{d}{dx}(xy) + \frac{d}{dx}y^2 = \frac{d}{dx}3 \\
\]

Answer 4

\[ \frac{d}{dx}x^2 = 2x \frac{dx}{dx} = 2x \]
\[ \frac{d}{dx}y^2 = 2y \frac{dy}{dx} \]

Answer 5

the derivative of xy uses the product rule
\[ \frac{d}{dx}(xy) = x \frac{d}{dx}y + y \frac{d}{dx}x =x \frac{dy}{dx} +y\]
and of course d/dy 3 is zero

solve for dy/dx (it is easier to first sub in 1 for x and y, because you want dy/dx at the point (1,1) ) to find the slope of the line tangent to the curve at that point.

finally, use
y - 1 = m(x - 1) for the equation of the line. m is dy/dx evaluated at (1,1)

Answer 6

ohh okay...

so m=0 in this problem?

so y-1=0(x-1) ?

Answer 7

\[ \frac{d}{dx}\left( x^2 + xy + y^2 = 3\right) \\
\frac{d}{dx}x^2 + \frac{d}{dx}(xy) + \frac{d}{dx}y^2 = \frac{d}{dx}3 \\
2x + x \frac{dy}{dx} + y + 2 y \frac{dy}{dx}=0
\]
treat dy/dx as a variable, and solve for dy/dx
if we first sub in 1 for x and y (because we are evaluating this at (1,1) )
\[ 2 + \frac{dy}{dx}+1 +2\frac{dy}{dx}=0 \]
what do you get for dy/dx ?

Answer 8

umm 5 dy/dx=0 ? dy/dx=0 ?

Answer 9

Have you reached the limits of your talents? how about rename dy/dx as X , so the equation is
2+X + 1 + 2X =0
collect like terms.

Answer 10

ohh okay..

so 3+3x=0 ?

Answer 11

3x=3?
x=1?

Answer 12

mucho bettero

Answer 13

except I get -1

Answer 14

oh right haha oops :P

yes x=-1 :)

so now we solve for the equation?

y-1=-1(x-1)
=y-1= -x +1
y= -x + 2 ?

Answer 15

yes, that looks good.

Answer 16

ah yay!! thank you!! :)

Answer 17

If you are losing concentration on this stuff, you might take a breather. It takes concentration to remember and apply your knowledge.

Answer 18

okay :) will do!! thank you1!!