Derivatives

1. Derivatives By: Jenn Gulya

2. The derivative of a function f with respect to the variable is the function f ‘ whose value at x, if the limit exists, is: This value is, also, representative of the slope of the function at a point.

3. When f'(a) doesn't exist. Where a is an x value, F(a) must be differentiable for f’(a) to exist. So at some points a derivative may not exist. (An example would be the absolute value graph, which lacks a derivative at x=0).

4. 1. Derivative of a Constant Function: If f is the function with a constant value c, then f’ = 0 2. Power Rule for Positive/ Negative Integer Powers of x: If n is a positive/negative integer, then Rules of Differentiation

5. 3. The Constant Multiple Rule If u is a differentiable function of x and c is a constant, then 4. The Sum and Difference Rule If u and v are differentiable function of x, then their sum and difference are differentiable at every point where u and v are differentiable. At such point

6. 5.The Product Rule The product of two differentiable functions u and v is differentiable, and 6. The Quotient Rule At a point where v 0, the quotient y= u/v of two differentiable functions is differentiable, and

7. Example 1: Y= x3+6x2-(5/3)x+16 Applying Rules 1 through 4, differentiate the polynomial term-by-term

8. Graph of original Function: It can be seen that when the original curve levels off and changes direction the graph of f’ crosses the x-axis. Graph of derivative:

9. Example 2: Calculations can be graphically supported. Differentiate f(x)=

10. When the calculated derivative and NDER of the function on the calculator are graphed together. They appear to be identical, which is a strong indication that the calculation is correct.

11. Jenn Gulya Will be majoring in Biology at Rutgers University in the Fall.