1. 7. Implicit Differentiation

2. Implicit Functions
All the methods we have covered so far have had y solved in terms of x, for example y = x3+1
What if instead we have an equation like
y4 + xy = x3 – x + 2
In this case, we say y is defined implicitly

3. Implicit Differentiation
Used when an equation cannot easily be solved for y
Steps
Find dy/dx of both sides of the equation (in other words, the derivative with respect to x for both sides of the equation)
Whenever there is a y, you must put a dy/dx along with the derivative
Solve for dy/dx

4. Example 1

5. What if instead this said
Example 2
What if instead this said

6. Find the tangent line to the ellipse below at (-1,2)
Example 3
Find the tangent line to the ellipse below at (-1,2)

7. Horizontal and Vertical Tangents
When the derivative is a ratio of 2 variable expressions
Horizontal tangents occur when numerator = 0
Vertical tangents occur when denominator = 0
No tangent exists when both = 0

8. Example 4
Find the coordinates where the graph of has horizontal and vertical tangents. At what point is the slope ¾?

9. Example 5
Find the coordinates where the graph of has horizontal and vertical tangents.

10. Example 6
Find dy/dx if

11. Finding a 2nd derivative implicitly Take 2nd derivative and plug in y’ Example 7