Chapter 3 Derivatives and Differentials 3.2 The rules for find derivative of a function

New Words Addition 加 Sum 和 Subtraction 减 Difference 差 Multiplication 乘 Product 积 Division 除 Quotient 商 A composite function 复合函数 Differentiate 求导

This section develops methods for finding derivatives of functions. As we known, functions can be built up from simpler functions by addition, subtraction, multiplication, and division. Thus, we first give the formula of derivatives of sum, difference, product, and quotient of two functions. 1.The Derivatives of the Sum, Difference, Product and Quotient Theorem 1

Proof To prove this theorem we must go back to the definition of the derivative.

(3) is a special case of the formula (2)

Note This completes the proof of theorem 1.

(3) Theorem 1 can extend to any finite number of derivable functions. The following examples use the formulas for the derivative of the sum, the difference, the product and the quotient. Example 1 SolutionBy the formula (1) of theorem 1

Example 2 SolutionBy the formula (4) of theorem 1

Example 3 SolutionBy the formula (4) of theorem 1

Example 4 Solution By the formulas of theorem 1

Using the definition of derivatives, we have at x=0 Example 5 Solution

2.The Rules for Finding Derivatives of Inverse Functions Theorem 2

Proof

Example 6 Solution

Example 7 Solution

3.The Derivative of a Composite Function (The Chain Rule) This section presents the most important technique for finding the derivative of a function. It turns out that we can easily compute the derivative of a composite function if we know the derivatives of the functions from which it is composed. Theorem 2 (The chain rule)

Proof

Note

The following examples apply the chain rule Example 8 Solution

Example 9 Solution

Example 10 Solution

Example 11 Solution

Example 12 Solution

Example 13 Solution

Example 14 Solution

Example 15 Solution

As these examples suggest, the chain rule is one of the most frequently used tools in the computation of derivatives.

4.The Derivatives of Elementary Functions a. The formulas of derivatives for the fundamental elementary function

b. The rules of operations for derivatives

Example 16 Solution

Example 17 Solution

Example 18 Solution We developed a formula called the chain rule for differentiating composite functions. The chain rule is the most commonly rule for differentiating functions.