# 5.1 Derivatives of Exponential Functions 5.1 DERIVATIVES OF EXPONENTIAL FUNCTIONS By: Rafal, Paola , Mujahid RECALL: y=ex (exponential function) LOGARITHM FUNCTION IS THE

INVERSE EX1: y=log4 x Therefore y=ex y=4x y=log e x 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 Eulers number or the natural number, where e= 2.718

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 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. 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 ) 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 Example 2: Find the derivative and simplify a)

f(x)= X2e2x f(x)= 2(x)(e2x ) + (X2 ) (2)(e2x ) use product rule f(x)= 2x e2x + 2 X2 e2x simplify terms f(x)=2x(1+x) e2x factor out 2x b) f(x)= ex x

f(x)= (1) (ex )(x) (ex )(1) X2 use quotient rule f(x)= x ex - ex X2 simplify terms f(x)= (x-1) (ex ) factor out ex ______ X2 Example 3: Determine the equation of the line tangent to f(x) = exx2 , where x= 2. Solution: Use the derivative to determine the slope of the required tangent.

f(x) = exx2 f(x)=x-2ex Rewrite as a product f(x)= (-2x-3)ex + x-2 (1)ex Product rule f(x)= -2ex x3 + ex x2 Determine common denominator f(x)=-2ex x3 + xex x3 f(x)= -2ex + xex x3 f(x)= (-2 + x) ex x3 Simplify Factor 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 Example 4: The number, N, of bacteria in a culture at time t, in hours, is N(t)=1000(30+e-). a) b)

c) 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?

## Recently Viewed Presentations

• Arial Calibri Default Design 1_Default Design Byzantine Art 6th-7th Centuries Byzantine Painting PowerPoint Presentation PowerPoint Presentation PowerPoint Presentation PowerPoint Presentation Byzantine Architecture St. Basil's St. Mark's PowerPoint Presentation PowerPoint Presentation Jigsaw PowerPoint Presentation Goreme Hosios Loukas ...
• The internet says… Defined as a main idea or an underlying meaning of a literary work that may be stated directly or indirectly . A work's thematic concept is what readers "think the work is about" . Its
• Division existed between the revivalists, known as New Lights, and the rationalist clergy, or Old Lights, who dominated the Anglican and Presbyterian churches. Great Awakening. The "Great Awakening" or "revival" had opened unprecedented splits in American Protestantism.
• The government is responsible for setting the target milestone(s) for the analysis, but the SRA effort is overseen by the contractor PM who generally sets the target probability. The bulk of work associated with an SRA is performed by the...
• photopeach.com. 12. Voki. Create your own avatar for free using their easy tools and sound access. Check out mine at . Voki.com. 13. Fix8. Software that allows you to animate yourself. Need a camera and webcam. ... Notable Web 2.0...
• Which of the cueing systems is the student using? Do a running record if you are not sure! Older students that struggle usually are over-dependent on phonics (visual); teach them to utilize a variety of systems.
• Napoleon's Empire Collapses. All of the main powers of Europe joined forces against Napoleon. 1813 - Russia, Britain, Prussia, Austria, Sweden defeat Napoleon in the Battle of Leipizg. Napoleon exiled to island of Elba, king restored to power in France...
• Describing Families. Module 3. Module 3. Describing Families (Date of slide or last update given at the top of the notes for each slide) Recognition of the family as an entity that can be recognized in descriptive cataloging is new...