1. Calculus and Analytical Geometry
MTH 104
Lecture # 10
Calculus and Analytical Geometry

2. Explicit and Implicit Functions
An equation of the form
is said to define
. For example
explicitly as a function of
Implicit Function

3. Example
Use implicit differentiation to find if
Solution

4. Implicit differentiation
To find
Differentiate both sides with respect to x, creating y as a differentiable function of x
Solve for
Example
Find of

5. Example
Use implicit differentiation to find of
Solution
Again differentiating both side implicitly

8. Example
Find the slopes of the tangent line to the curve
at the points (2, -1) and (2, 1).
Solution

9. Slope of the tangent line at (2, -1)

10. Derivatives of logarithmic functions
Generalized derivative formulas

11. Examples
Find
Solution 1.

12. 2.
Product rule 3.

13. Product rule

14. 4. 5.
Example Find
using logarithmic differentiation

15. Solution
Taking ln of both sides
Differentiating with respect to

17. Derivatives of exponential
Generalized form

18. Examples 1. 2. 3. 4.

19. Derivatives of inverse Trigonometric Functions
Generalized derivative formulas 1. 2. 3. 4. 5.

20. 6.
Example Find if
solution

21. Example Find if
solution