1. Mathematics Medicine 2010-2011 The Derived Function

2. Given a function y = f(x). Definition f(x) is said to have a derived function at x if and only if the following limit exists and is finite; the function defined by the limit is called the derived function of f(x): The derived function is also called the derivative of f(x) and the process of finding the derivative is called differentiation. A function f(x) which has a derivative at a point x = a is said to be differentiable at x = a.

3. Picture x=ax=a+h A B D E C Y X f(a+h)-f(a) y=f(x) Secant line Tangent line

4. Geometric Interpretation of the Derived Function Let have a graph (see the picture). Let x = a be a point for which The y coordinate of point B is. Let h be a real number, either positive or negative. and Notice that the length of the segment BE is exists and letexists. Consequently, is difference in the y coordinates of points D and B Then a + h is a point on the x axis and is the y coordinate of point D. and may be positive or negative. Its absolute value is depicted in the diagram as DE, the difference in the lengths of

5. The ratio is slope of the line determined by B and D. Such a line is a secant line. Now think of h taking on values close to 0. For each h the ratio is the slope of the corresponding secant line. When h is are very close to 0, the points B and D are each on the graph of and are very close together Consequently, for small h, the secant line through B and D is very close to the line tangent to at x = a

6. Since the secant lines through B approach the line tangent to f(x) at x = a, the slopes of the secant lines approach the slope of the tangent line. Since the slope of each secant line is given by the slope of the tangent is given by That is by the derived function of Geometric Interpretation of is the slope of the line tangent to at x = a evaluated at x = a

7. Rules for Finding f'(x) The derived function for most functions can be written down by application of simple rules. Theorem 1: Theorem 2: Theorem 3: If, k a constant, then The derived function of the product of a constant k and a function is the product of k and If where m is a positive integer, then

8. Rules Theorem 4: Theorem 5: Ifandexists for all n then If then

9. Rules Theorem 6: Theorem 7: Theorem 8: If then If then If, m a positive integer,, then

10. Rules Theorem 9: Theorem 10: If then If then Theorem 11: If then Theorem 12: If then

11. Rules Theorem 13: Theorem 14: If then If then Theorem 15: If then Theorem 16: If then

12. Rules Theorem 17: Theorem 18: If then If then Theorem 19: If then Theorem 20: If then

13. Rules Theorem 11: Theorem 12: If where r and s are differentiable function, then If where r and s are differentiable function and, then

14. Rules Theorem 13: If (means, where and are differentiable function, then