1. 4.1 Implicit Differentiation 4.1.1 Definition. We will say that a given equation in x and y defines the function f implicitly if the graph of y = f(x) coincides with a portion of the graph of the equation. Example: The equation implicitly defines functions The equation implicitly defines the functions

2. There are two methods to differentiate the functions defined implicitly by the equation. For example: One way is to rewrite this equation as, from which it follows that Two differentiable methods

3. With this approach we obtain The other method is to differentiate both sides of the equation before solving for y in terms of x, treating y as a differentiable function of x. The method is called implicit differentiation. Since, Two differentiable methods

4. Example: Use implicit differentiation to find dy / dx if Solution: Example

5. Example: Find dy / dx if Solution: Example

6. By implicit differentiation, we can show that if r is a rational number, then Example: Example

7. In general, Example

8. 4.2 Derivatives of Logarithmic Functions Generalized derivative formulas

9. Example Solution: Example

10. Example: Solution: Example

11. From section 4.1, we know that the differentiation formula holds for rational values of r. In fact, we can use logarithmic differentiation to show that holds for any real number (rational or irrational). Example:

12. 4.3 Derivatives of Exponential and Inverse Trigonometric Functions Differentiability of Exponential Functions Example:

13. Derivatives of the Inverse Trigonometric Functions

14. Example Example: Find dy/dx if Solution: Example: Find dy/dx if Solution:

15. 4.4 L’Hopital’s Rule; Indeterminate Forms

16. Applying L’hopital’s Rule

17. Example: Find the limit using L’Hopital’s rule, and check the result by factoring. Solution: The numerator and denominator have a limit of 0, so the limit is an indeterminate form of type 0/0. Applying L’Hopital’s rule yields This agrees with the computation Example

18. Example: Find Solution: The limit is a indeterminate form of type 0/0. Applying L’Hopital’s rule yields Example

19. Example: Find Solution: The limit is a indeterminate form of type 0/0. Applying L’Hopital’s rule yields Example

20. Indeterminate Forms of Type  / 

21. Example: Find Solution: The limit is a indeterminate form of type Applying L’Hopital’s rule yields In fact, we can use LHopital’s rule to show that Example

22. Example: Find Solution: The limit is a indeterminate form of type Applying L’Hopital’s rule yields Similar methods can be used to find the limit of f(x)/g(x) is an Indeterminate form of the types: Example