1-Basic Rules of Differddddentiation.pptx

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1. DERIVATIVES BASIC RULES OF DIFFERENTIATION

2. Derivative of a function at a certain point, is the slope of the function at that particular point. In the previous lesson, we derived the formula for the derivatives as: Using this formula can be cumbersome. If this limit exist, for certain values of x in the domain of function f , we say that the functions is differentiable at x .   DERIVATIVE OF A FUNCTION

3. In this lesson we are going to learn and use some basic rules of differentiation that are derived from the definition. For these rules, let’s assume that we are discussing differentiable functions. BASIC RULES OF DIFFERENTIATION

4. The common notations for derivatives are:           NOTATIONS FOR DERIVATIVES

5. This rule states that the derivative of a constant is zero. For example, for any constant, c.       RULE 1: DERIVATIVE OF A CONSTANT

6.   , where n is any real number This rule states that the derivative of x raised to a power is the power times x raised to a power one less or n – 1. For example,     Notice that the derivative is the original power, 5 times x raised to the fourth, which is one less than 5. RULE 2: THE POWER RULE

7.   , where c is a constant This rule states that the derivative of a constant times a function is the constant times the derivative of the function. For example, find the derivative of     RULE 3: DERIVATIVE OF A CONSTANT MULTIPLE OF A FUNCTION

8. RULE 4: DERIVATIVE OF A SUM OR DIFFERENCE   This rule states that the derivative of a sum or difference is the sum or difference of the derivatives. For example, find the derivative of x 2 + 2 x - 3   The derivative of x squared is done by the Power Rule (2), the derivative of 2x is done by rule 3 and power rule and the derivative of 3, a constant is 0.

9. More Examples: Find   You’ll notice none of the basic rules specifically mention radicals, so you should convert the radical to its exponential form, x 1/2 and then use the power rule.

10.   Rewrite the expression so that you can use the basic rules of differentiation.   Now differentiate using the basic rules.   More Examples: Find

11.   RULE 5: PRODUCT RULE In other words: The derivative of f times g is the first times the derivative of the second plus the second times the derivative of the first.

12. Example:     PRODUCT RULE     Derivative of the first function Derivative of the second function

13. Another notation to express the product rule:   Some times the two functions are expressed as u and v , so here u is u(x) and v is v(x) meaning both are functions of x. This is sometimes memorized as:                        

14. 2. Same derivative by expanding and using the Power Rule.   1. The Product Rule     Find the derivative of  

15. EXAMPLE 2 OF THE PRODUCT RULE: Find f’(x) for    

16. Recall when you are multiplying the same base you add the exponents.   Example 2 continued…

17. Find the derivative of the function below using the product rule of differentiation or the power rule.