1. 2.8 Implicit Differentiation
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. Two differentiable methods
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

3. Two differentiable methods
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.
With this approach we obtain
Since ,

4. Implicit Differentiation

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

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

7. 2.10 Logarithmic Functions

8. Logarithm Function with Base a

9. Natural Logarithm Function
Logarithms with base e and base 10 are so important in applications that
Calculators have special keys for them.
logex is written as lnx
log10x is written as logx
The function y=lnx is called the natural logarithm function, and y=logx is
Often called the common logarithm function.

10. Properties of Logarithms

11. Properties of ax and logax

12. Derivative of the Natural Logarithm Function
Note:
Example:
Solution:

13. Example
Example:
Solution:

14. Derivatives of au
Note that
Example:

15. Derivatives of logau
Note that
Example:

16. The Number e as a Limit

17. 2.11 Inverse Trigonometric Functions
The six basic trigonometric functions are not one-to-one (their values
Repeat periodically). However, we can restrict their domains to intervals
on which they are one-to-one.

19. Six Inverse Trigonometric Functions
Since the restricted functions are now one-to-one, they have inverse, which we denoted by
These equations are read “y equals the arcsine of x” or y equals arcsin x” and so on.
Caution: The -1 in the expressions for the inverse means “inverse.” It does
Not mean reciprocal. The reciprocal of sinx is (sinx)-1=1/sinx=cscx.

22. Derivative of y = sin-1x
Example: Find dy/dx if
Solution:

23. Derivative of y = tan-1x
Example: Find dy/dx if
Solution:

24. Derivative of y = sec-1x
Example: Find dy/dx if
Solution:

25. Derivative of the other Three
There is a much easier way to find the other three inverse trigonometric
Functions-arccosine, arccotantent, and arccosecant, due to the following
Identities:
It follows easily that the derivatives of the inverse cofunctions are the negatives
of the derivatives of the corresponding inverse functions.