1. Implicit Differentiation

2. This is not a function, but it would still be nice to be able to find the slope.
Note use of chain rule.

3. This can’t be solved for y.
This technique is called implicit differentiation.
1 Differentiate both sides w.r.t. x.
2 Solve for

4. Find the equations of the lines tangent and normal to the curve at .
We need the slope. Since we can’t solve for y, we use implicit differentiation to solve for
Note product rule.

5. Find the equations of the lines tangent and normal to the curve at .

6. Examples: x2y3 – xy = 10 x2+3xy+y2 = -1 x2+y2 = 25 y3-x2 = x + y
3y + ln y = 4ex
x2y= y3x + 5y + x
25x2 + 8x – 16y2 - 4y – 9 = 0

7. More examples: Find dy/dx
y = sin x + cos y
x2y + y3 + 3 = xy
xy3 + xy = 6
ex + cos y = ln y6
x2 + 2xy + y4 = 1
x2y2 – ex + 2y = 4
sin x/y = ½
cos (x+y) + sin (x+y) = 1/3
ecos x + esin y = 1/4