Section 2.4 Developing Derivative Properties
Standards
- Calculate polynomial derivatives
- Calculate trigonometric derivatives
- Calculate exponential derivatives
- Calculate derivatives with the product rule
- Calculate derivatives with the quotient rule
Just like with limits we can calculate derivatives by breaking them down using arithmetic and algebraic properties.
Our emphasis will be on calculating derivatives, though two of the rules make important statements about the relationship between rates in applications. Calculating derivatives is a skill useful in recognizing relationships in other classes, though because of computer it is no longer used in jobs.
Subsection 2.4.1 Calculating Derivatives Using the Definition
The definition derives from the concepts. Using it to calculate derivatives takes some creativity.
Example 2.4.1. Calculate the derivative of
Example 2.4.2. Calculate the derivative of .
Solution.
Example 2.4.3. Calculate the derivative of .
Calculate the derivative of
Solution.
Subsection 2.4.2 Arithmetic Derivative Properties
Subsubsection 2.4.2.1 Derivative of a Sum
Subsubsection 2.4.2.2 Derivative of a Scalar Product
Subsubsection 2.4.2.3 Derivative of a Product
There is also a property for a derivative of a product, but it is not simply applying the limit property for a product. Provide a reason for each line of this proof.
We can illustrate the produce rule using a product to represent an area of a rectangle as in Figure 2.4.4 Note that the change in the area of the rectangle is the gnomon (inverted L shape) consisting of and This matches the product rule except for the small square In the limit this square’s area goes to zero. Indeed in early development of the derivative this is exactly how they did it.
Instructions.
Use the slide labeled da to view the limit as it approaches zero giving us the derivative.
Subsubsection 2.4.2.4 Derivative of a Quotient
Subsection 2.4.3 Using Properties to Calculate Derivatives
Subsubsection 2.4.3.1 Derivative of Integer Powers
Checkpoint 2.4.5.
Using the product rule calculate the derivatives of the following functions. Conjecture the pattern for all integer powers. Explain why the product rule cannot be used to prove this pattern for
Subsubsection 2.4.3.2 Derivatives of trigonometric functions
Checkpoint 2.4.6.
Subsubsection 2.4.3.3 Calculating Using Multiple Properties
Many functions involve more than one of the properties. This leads to the question about how to apply more than one property when calculating a derivative.
Example 2.4.7.
Checkpoint 2.4.8.
Why was the sum property used before the product property in Example 2.4.7? What order is this?