Chapter 3 – Differentiation Rules

1. Chapter 3 – Differentiation Rules 3.5 Implicit Differentiation Section 3.5 Implicit Differentiation

2. Implicit Differentiation • In some cases it is possible to solve such an equation for y as an explicit function (or several functions) of x. Section 3.5 Implicit Differentiation

3. Implicit Differentiation • For instance, if we solve the equation for y, we get , so two of the functions determined by the implicit equation are and . Section 3.5 Implicit Differentiation

4. Implicit Differentiation • In some cases it is not possible to solve such an equation for y as an explicit function (or several functions) of x. Section 3.5 Implicit Differentiation

5. Implicit Differentiation We don’t need to solve an equation for y in terms of x in order to find the derivative of y. Instead we can use implicit differentiation. This consists of differentiating both sides of the equation with respect to x and then solving for dy/dx. Section 3.5 Implicit Differentiation

6. Example 1 Find the derivative using implicit differentiation. Section 3.5 Implicit Differentiation

7. Example 2 Use implicit differentiation to find the second derivative of Section 3.5 Implicit Differentiation

8. Example 3 • Use implicit differentiation to find dy/dx for the Folium of Descartes • Find an equation for the tangent line to the Folium of Descartes at the point (3/2, 3/2). • At what point(s) in the first quadrant is the tangent line to the Folium of Descartes horizontal? Section 3.5 Implicit Differentiation

9. Try these… Section 3.5 Implicit Differentiation

10. Derivatives of the Inverse Trig Functions Section 3.3 Derivatives of Trig Functions

11. Example 4 Find the derivative and simplify where possible. Section 3.5 Implicit Differentiation

12. Try these… Section 3.5 Implicit Differentiation