1. Lesson: ____ Section: 3.7  y is an “explicitly defined” function of x.  y is an “implicit” function of x  “The output is …”

2. Consider the circle as a whole. It is still a curve with an equation and this curve has a tangent line at each point. The slope of the tan line can be found by Differentiating both sides of the equation with respect to x. Chain Rule! Note that the derivative depends on both x and y rather than just x. Discuss the polarity of the slope of this circle in the second quadrant. Does this agree with the derivative? How about QIII? This works for all points on the circle excepts where a vertical tangent line exists such as (-2.0) or (2,0) …so the idea of a derivative still makes sense

3. Ex.1 Make a table of x and approximate y values for the equation near (7,2). Chain Rule & Product Rule Since we can’t solve for y in terms of x, let’s find an equation for the tangent line and use that tangent line to approximate other points near (7,2). When x=7 and y=2 Tangent line at (7,2)

4. Notice that in a small neighborhood around (7,2), the tangent line gives a good approximation of the curve. xy 6.8 6.9 7 7.1 7.2 1.92 1.96 2 2.04 2.08 Find the coordinates of some other points near (7,2) Finding other points on the curve by plugging into the original equation is a long and cumbersome process.

5. As this can’t be solved for y, this is, once again, an implicit differentiation question. implicit differentiation: 1 Differentiate both sides wrt x. 2 Solve for.

6. Check out the graph! We can now use dy/dx to find the slope of the curve at any point on this crazy thing.

7. Find the equations of the lines tangent and normal to the curve at. Note product rule. Find the equations of the lines tangent and normal to the curve at.

8. tangent:normal:

11. Higher Order Derivatives Find if. Substitute back into the equation.

12. Another crazy looking graph just for fun