2.5 Derivatives of Polynomials 1 The four-step process has been introduced to found derivatives. In the present and the next lecture, using this process, we develop differentiation rules to directly apply in the future. Constant Rule: If c is a constant, then Proof: The given function is y=c. We apply the four-step process to this function: Step 1. y=f(x)=c, y+  y=f(x  x  c. Step 2.  y=c - c=0. Step 3.  y/  x = 0. Step 4. Picture: the function is a horizontal line, and its slope at any point is 0.

2 y=x is a diagonal line. So, its slope is 1: dx/dx = 1. Prove! Power Rule: If n>0, Proof (for integer n>0 only): Again, we apply the four-step process to this function: Step 1. Step 2. Step 3. Step 4.

3 Sum Rule: For any functions of x, u(x) and v(x), Proof: Again, we apply the four-step process to the sum: Step 1. Denote u=f(x), v=g(x), y=h(x)=u+v=f(x)+g(x), then y+  y = h(x  x  f(x+  x)+g(x+  x). Step 2.  y = h(x+  x)-h(x) = f(x+  x)+g(x+  x)-f(x)-g(x). Denote  u=f(x+  x)-f(x),  v=g(x+  x)-g(x). Then,  y=  u+  v Step 3. Step 4. Using the theorem that the limit of a sum is equal to the sum of limits (Sec. 2.2, p. 64), we state

4 Constant Multiplier Rule: For any constant c and function u(x), Prove the above statement doing steps 1 through 4. At the last step, you need to apply another part of the theorem from Sec. 2.2 (p.64): For any constant c, Example: Differentiate

5 Exercises: Differentiate

6 Homework Section 2.5: 3,9,13,17,19,21,23,27.