1 / 12

5.1 DERIVATIVES OF EXPONENTIAL FUNCTIONS

5.1 DERIVATIVES OF EXPONENTIAL FUNCTIONS. By: Rafal, Paola , Mujahid. RECALL:. y=e x (exponential function) LOGARITHM FUNCTION IS THE INVERSE EX1: y=log 4 x y=4 x Therefore y=e x y=log e x. IN THIS SECTION:.

teva
Download Presentation

5.1 DERIVATIVES OF EXPONENTIAL FUNCTIONS

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. 5.1 DERIVATIVES OF EXPONENTIAL FUNCTIONS By: Rafal, Paola , Mujahid

  2. RECALL: • y=ex (exponential function) • LOGARITHM FUNCTION IS THE INVERSE • EX1: y=log4 x y=4x • Therefore y=ex y=log e x

  3. IN THIS SECTION: • The values of the derivative f’(x) are the same as those of the original function y=ex • THE FUNCTION IS ITS OWN DERIVATIVE f(x)=ex f’(x)=ex • “e” is a constant called Euler’s number or the natural number, where e= 2.718

  4. The product, quotient, and chain rules can apply to exponential functions when solving for the derivative. f(x)=e g(x) Derivative of composite function: f(x)=e g(x) f’(x)= e g(x) g’(x) by using the chain rule

  5. Key Points • f(x)= ex , f’(x)= ex • Therefore, y= ex has a derivative equal to itself and is the only function that has this property. • The inverse function of y=ln x is the exponential function defined by y= ex.

  6. Example 1: Find the derivative of the following functions. a) y= e 3x+2 b) y= ex2+4x-1 y’= g’(x) (f(x)) y’= g’(x) (f(x)) y’=(3)(e 3x+2 ) y’= (2x+4)(ex2+4x-1 ) y’= 3 e 3x+2 y’= 2(x+2)(ex2+4x-1 )

  7. You can also use the product rule and quotient rule when appropriate to solve. • Recall: Product rule: f’(x)= p’(x)(q(x)) + p(x)(q’(x)) Quotient rule: f’(x)=p’(x)(q(x)) – p(x)(q’(x)) ___________________________ q(x)2

  8. Example 2: Find the derivative and simplify • f(x)= X2e2x f’(x)= 2(x)(e2x ) + (X2 ) (2)(e2x ) use product rule f’(x)= 2x e2x + 2 X2 e2xsimplify terms f’(x)=2x(1+x) e2x factor out 2x

  9. f(x)= ex ÷ x f’(x)= (1) (ex )(x) – (ex )(1) ÷ X2use quotient rule f’(x)= x ex - ex ÷ X2simplify terms f’(x)= (x-1) (ex ) factor out ex ______ X2

  10. Example 3: Determine the equation of the line tangent to f(x) = ex÷x2 , where x= 2. Solution: Use the derivative to determine the slope of the required tangent. f(x) = ex÷x2 f(x)=x-2exRewrite as a product f’(x)= (-2x-3)ex+ x-2 (1)exProduct rule f’(x)= -2ex ÷ x3 + ex÷ x2Determine common denominator

  11. f’(x)=-2ex ÷ x3 + xex÷ x3Simplify f’(x)= -2ex + xex÷ x3Factor f’(x)= (-2+ x) ex÷ x3 When x=2, f(2) = e2÷ 4. When x=2, f’(2)=0 . so the tangent is horizontal because f’(2)=0. Therefore, the equation of the tangent is f(2) = e2÷ 4

  12. Example 4: The number, N, of bacteria in a culture at time t, in hours, is N(t)=1000(30+e-). • What is the initial number of bacteria in the culture? • Determine the rate of change in the number of bacteria at time t. • What is happening to the number of bacteria in the culture as time passes?

More Related