Definition of the Derivative

Transcription

1 Te Limit Definition of te Derivative Tis Handout will: Define te limit grapically and algebraically Discuss, in detail, specific features of te definition of te derivative Provide a general strategy of finding te derivative by definition Demonstrate te process of finding te derivative by definition using four distinct examples. Te Derivative Tis section will: Introduce te grapical limit of a derivative Define te derivative algebraically as a limit Provide strategies for andling specific features of te algebraic definition Te derivative is te primary topic of calculus I. Te derivative is te instantaneous rate of cange of a function at any point. Grapically, te derivative is te slope of te tangent line troug te point. Te tangent line is a line tat passes troug two infinitesimally close points on a curve. Tis can be tougt of as just toucing te curve at a point. Tis is pictured below. More information on te grapical definition of te derivative can be found in 3. Tangents and te Derivative at a Point in Tomas Calculus. Algebraically, te derivative can be found by taking a particular limit, called te limit definition of te derivative: fx + ) fx) Definition of te Derivative Were f x) is te derivative, fx) is te function we wis to differentiate, and is a dummy variable tat will not appear in te final result. Image source: ttp:// Differentiate means to find te derivative of. Differentiation is te process of finding a derivative.

2 Te Limit Definition of te Derivative Tat is, we can find any derivative by substituting te function into te Definition of te Derivative and evaluating te limit as approaces zero. Practically, to evaluate te limit we will first evaluate fx + ). For example, if fx) = x + 4. We may evaluate fx + ) if we treat x + ) as a quantity tat may be substituted into te function tat is, werever we replace every x wit x + ). For tis function, fx + ) would be: fx + ) = x + ) + 4 See Determining fx + ) for more examples on tis process. After we evaluate fx + ), we ten manipulate te resulting expression until we may substitute in = witout obtaining a form. For example, if Substituting in = evaluates to x + x expression algebraically. Simplifying x + x x f x) x + x x + ) + =, so we need to manipulate simplify ) te + = x + = x + Te limit f x) x + ) may be evaluated by directly substituting in = to obtain x. Do Problem on practice problems for more practice on tis process. Summary: Wen evaluating te derivative: First evaluate fx + ) so tat it may be substituted into te definition of te derivative Manipulate te resulting expression i.e. from te definition of te derivative) until you may substitute = witout obtaining a form. Determining fx + ) Tis section will provide tree more examples on evaluating expressions of te form fx + ) for tese following functions: f x) = x 3 x f x) = x + 3 f 3 x) = x

3 Te Limit Definition of te Derivative To evaluate, we treat x + ) as a quantity and place x + ) directly in te function in place of x, as follows: f x + ) = x + ) 3 x + ) f x + ) = x + ) + 3 f 3 x + ) = x + ) Finding te Derivative: Tis section will Provide a general strategy for finding derivatives by definition. Provide specific strategies for four distinct worked examples. Examples are te best way to learn ow to take derivatives by definition. Te following four examples will illustrate tis process of finding te derivative. Te general process of eac example is: Start wit te definition of te derivative Explicitly write out fx + ) and fx) Manipulate te expression algebraically simplify ). Tis is te step tat canges te most from example-to-example and eac example will ave its own algebraic manipulate. Continue to manipulate te algebra until we may substitute in = witout obtaining a form. Substitute in =. Simplify te result as muc as possible. We will find te derivatives of tese following functions: Example fx) = x + 4 Page 4) Example fx) = x+3 Page 5) Example 3 fx) = x + 5 Page 6) Example 4 fx) = Page 7-8) x Tese examples are cosen to provide as muc diversity as possible. Example is cosen to illustrate te metod, examples -4 are cosen to illustrate specific metods of algebraic manipulation. 3

4 Te Limit Definition of te Derivative Example : We will find te derivative of fx) = x + 4. Write te definition of te derivative: fx + ) fx) Substitute in our functions for fx + ) and fx): Expand te factors in te numerator: Combine like terms in te numerator. Factor out te in te numerator. x + ) + 4) x + 4) x + x x 4 x + x + ) Cancel out te in te numerator and denominator. x + Substitute in =, as we may substitute = witout obtaining a form Tis gives us our final result: te derivative: f x) = x + f x) = x 4

5 Example : Simple Rational Function 3 Let Campus Academic Resource Program Te Limit Definition of te Derivative fx) = x + 3 To find te derivative, substitute fx) into te definition of te derivative. Reduce te number of fractions into te expression x x + 3 x x + 3 = x x + 3 ) by factoring out. Tis is similar to ow we can write 3 = 3) or x = x, except our y y numerator in tis case as fractions. Our derivative is x x + 3 ). Simplify. To do tis, we want bot fractions to ave a common denominator and x++3 x+3 follow standard fraction aritmetic were a + c = ad+bc. If you are unclear on ow to add fractions, b d refer to te CARP andout on fractions 4. Simplifying bd x x + 3 x + 3 ) x + 3 x x )) x + 3 x + + 3)x + 3) x x + + 3)x + 3) ) + 3 x + + 3) x x + + 3)x + 3) ) x + + 3)x + 3) x + + 3)x + 3) In te last step, we may substitute in = witout leading to a form, and in doing so we obtain f x) = x + 3) 3 A rational function is defined as a quotient of two polynomial functions. Tis rational function is simple because te numerator is, wic is te simplest polynomial. 4 ttps://sites7.sfsu.edu/sites/sites7.sfsu.edu.carp/files/pdf/mat/algebrai-mat6/introductiontofractions.pdf 5

6 Te Limit Definition of te Derivative Example 3: Radical Expression fx) = x + 5 To find te derivative, substitute fx) into te definition of te derivative. x + ) + 5 x + 5 We want to rewrite te above expression witout square roots wic requires te concept of a conjugate. A conjugate of an expression means tat you take tat expression and cange a sign. For q p, te conjugate is q + p. Te product q p, ) q + p) = p + q by difference of squares, wic you may verify by using FOIL on te expression q p, ) q + p). We will refer to tis process as conjugation. Te conjugate of x + ) + 5 x + 5 is x + ) x + 5. To multiply te numerator by x + ) x + 5, also multiply te denominator by x + ) x + 5: x + ) + 5 x + 5 x + ) x + 5 ) x + ) x + 5 ) Te numerator is x + ) + 5 x + 5) by difference of squares. f x + ) + 5 x + 5) x) x + ) x + 5) x x 5 x + ) x + 5) x + ) x + 5) x + ) x + 5) Substitute in =, as you may do so witout obtaining a indeterminate form: f x) = x x + 5) = x + 5 = x + 5 6

7 Te Limit Definition of te Derivative Example 4: Mixing fractions and radicals. Let fx) = x Tecniques present in bot example and example 3 must be used and sould be reviewed and understood before proceeding wit tis example. Substitute fx) into te definition of te derivative: x + ) x Using te same tecnique in example, we may rewrite te previous expression as x + ) x ) Use conjugation to simplify te square roots. Te conjugate of Multiply: Simplify te product is +. x+) x x+) x x + ) x ) x + ) + x ) x + ) + x ) x+) x ) + x+) x ) using difference of squares: x + ) x ) x + ) + x ) We will not be able to simplify until te last step. Simplify ) by using x+) + x ) x+) x standard fraction aritmetic like tat used in example : x x + )) x) x + ) x + )) x) ) x + ) + x ) x x + )) x + )) x) ) x + ) + x ) Simplify as muc as possible: x + x + ) x + )) x) ) x + ) + x ) 7

8 Te Limit Definition of te Derivative x + x + x + )) x) ) x + ) + x ) x + )) x) ) x + ) + x ) x + )) x) ) x + ) + x ) Substitute in =, as we may do so witout obtaining a indeterminate forml f x) = x) x) x + x ) = x) x) x ) We may now simplify x ) ; x f x) = x) x) x ) x = = x x) x) x x) We may simplify te previous expression even furter by noting x = x), so te previous expression becomes f x) = x) x) We may apply laws of exponents 5 f x) = x) 3 = x) 3, wic is a form you obtain if you were to apply derivative rules. 5 Refer to te formula seet in front cover of calculus textbook for Laws of Exponents 8

9 Te Limit Definition of te Derivative Glossary: Tis section will define key terms used in tis section Algebraic Manipulation: Often called simplifying. Refers to te process of rewriting an algebraic expression in an equivalent expression tat is more manageable. E.g., x +x = x + is an example of algebraic manipulation. Conjugate: A quantity tat is similar to anoter quantity except for a sign cange. For tis andout, we deal wit expressions q + p, wic as a conjugate q p. Conjugation: Te process of simplifying an expression by multiplying it by its conjugate. Definition of Derivative: Also known as algebraic or limit definition of te derivative. Refers to te equation lim fx+) fx). Difference of Squares: Te identity tat says x y ) = x + y)x y). A particularly useful instance of tis identity for tis andout is x y = x + y) x y). Practice Problems: Problem : Wic of te following limits may be evaluated by substituting in = witout obtaining a form? Answers given at bottom of page) x x+) x ).) lim.) x+ lim x 3.) lim Problem : For eac of te following functions, write out fx + ).) fx) = x +.) fx) = x 3.) fx) = x+4 4.) fx) = x Answers for problem :.) No,.) Yes, 3.) No 9

10 Te Limit Definition of te Derivative For problems 3 to 6, find te derivative of te following problems. Problem 3: Find te derivate of fx) = x + Problem 4: Find te derivative of fx) = x

11 Te Limit Definition of te Derivative Problem 5: Find te derivative of fx) = x+4 Problem 6: Find te derivative of fx) = x

12 Te Limit Definition of te Derivative Citations: Tomas, G. B., Weir, M. D., Hass, J., & Giordano, F. R. ). Tomas' Calculus Early Transcendentals. Pearson.