1. Implicit Differentiation
Section 3.7
Implicit Differentiation

2. Example 1
Find dy/dx if 𝑦 2 =𝑥.

3. Example 2
Find dy/dx if 𝑥 2 𝑦+𝑥 𝑦 2 =6.

4. Example 3
Find the slope of the circle 𝑥 2 + 𝑦 2 =25 at the point (3, -4).

5. Example 4
Show that the slope dy/dx is defined at every point on the graph of 2𝑦= 𝑥 2 +𝑠𝑖𝑛𝑦.

6. Example 5
Differentiate.
𝑥𝑠𝑖𝑛𝑦+𝑦𝑐𝑜𝑠𝑥=1

7. Implicit Differentiation Process
Differentiate each side with respect to x.
Collect all y’ terms on one side.
Factor out y’.
Solve for y’.

8. Example 6
Find the tangent and normal to the ellipse 𝑥 2 −𝑥𝑦+ 𝑦 2 =7 at the point (-1, 2).

9. Example 7
Find d²y/dx² if 2 𝑥 3 −3 𝑦 2 =8.

10. TOTD
Find the lines that are (a) tangent and (b) normal to the curve at the given point (-1,0).
6 𝑥 2 +3𝑥𝑦+2 𝑦 2 +17𝑦−6=0

11. Example 8
Find the second derivative of 𝑥 2 + 𝑦 2 =1.

12. Example 9
Differentiate.
𝑦= 𝑥 2 3

13. Example 10
Differentiate.
𝑦=𝑐𝑜 𝑠 − 1 5 𝑥

14. TOTD
Find dy/dx
𝑦=𝑥 𝑥 2 +1