The document discusses derivatives of functions and inverse functions. It defines a function and inverse function, and explains that a function must be one-to-one to have an inverse. It also states that the slope of the tangent line to the inverse function at a point (a,b) is the inverse of the slope of the tangent line to the original function at the same point (b,a).

1. Derivatives of Functions and Inverse Functions E. Alexander Burt Potomac School

3. y=f(x)

4. The function may also take the form of a set of equations

5. The function may also take the form of a table of values.

7. y=3 x

8. y=5x-2

10. y=x 2 for -2<x<2

11. y=sin(x) for - p <x< p

12. These functions can be made into one to one functions by restricting the domain.

13. y=x 2 for 0<x<2

14. y=sin(x) for - p /2<x< p /2

19. Solve for x:

21. This is true of any point (a,b) on the function: (b,a) will lie on the inverse.

23. f'(x) is the derivative of f(x) and g'(x) is the derivative of g(x).

25. Or g'(f(a))=1/m

30. Therefore 1,3 is on g

32. Functions defined graphically An example of a function (orange) and its inverse (blue). Note that for every (a,b) pair on the function, there is a corresponding (b,a) pair on the inverse. y=x g(x) f(x) (4,1.5) (1.5,4)

35. If the graph of the function passes through (a,b) then the graph of the inverse passes through (b,a)

36. Using the above notation, the slope of the tangent line to the function at a is the reciprocal of the slope of the tangent line to the inverse at b.

37. Not exactly Haiku... The slope of the line tangent to the inverse of a function at (a,b) is the inverse of the slope of the line tangent to the function at (b,a)