1 3.3 Rules for Differentiation Badlands National Park, SD

2 Derivative of a Constant Function Consider the graph of the function: f(x) = c. What is the slope of the function at any given x? Therefore: The derivative of a constant equals zero.

3 Power Rule for Derivatives In previous examples given in class, you may have noticed that the degree of derivative is one lower than the degree of the function. Here is the rule that will allow you to shortcut derivatives of polynomials and other similar functions. Power Rule for Derivatives Lower the exponent by one Make the original exponent into the coefficient Warning This rule only works for functions that have variable bases and numbers as exponents, do not use on other types of functions!!!!

4 Find the derivative:

5 Sums & Differences of Differentiable Functions Derivatives may be done term by term in an expression. Example:

6 Find the derivative:

7 Second Derivatives What happens if you take the derivative of a derivative? The second derivative tells us how quickly the slope of the graph is changing. It is also the slope of the graph of the derivative.

8 Second Derivatives Find the second derivative of the function:

9 At what coordinates (x,y) for f(x) is the tangent line horizontal? Find where f ‘(x) = 0.

10 At what coordinates (x,y) for f(x) is the tangent line horizontal? Find where f ‘(x) = 0.

11 Differentiate the function.

12 Differentiate the function.

13 Differentiate the function.

14 Differentiate the function.

15 Differentiate the function.

16 Differentiate the function.

17 Differentiate the function.

18 Differentiate the function.

19 Differentiate the function.