A community for students.
Here's the question you clicked on:
 0 viewing
anonymous
 5 years ago
(another characterisation of y = e^x  it is the only exponential function whose gradient at its yintercept is 1)
(a) prove that y= a^x has derivative at y=a^x loga
(b) prove that y= a^x has gradient 1 at its y intercept if and only if a = e
(c) prove that y=Aa^x has gradient 1 at its y intercept if and only if a=e^(1/a)
anonymous
 5 years ago
(another characterisation of y = e^x  it is the only exponential function whose gradient at its yintercept is 1) (a) prove that y= a^x has derivative at y=a^x loga (b) prove that y= a^x has gradient 1 at its y intercept if and only if a = e (c) prove that y=Aa^x has gradient 1 at its y intercept if and only if a=e^(1/a)

This Question is Closed

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0f(x) = a^x f'(x) = [f(x+h)  f(x)]/h = [a^(x+h)  a^(x)]/h = [a^(x)a^(h)  a^(x)]/h = a^(x)[a^(h)1]/h lim h> 0 = a^x loga

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0is that a general formula for intergrating exponentials?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0@and: differentiating.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0its the fundamental process for evaluating derivatives..

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0oh sorry yeah i ment diffrentiating

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0the part where ive replaced (a^h  1 )/ h as loga where h tends to zero is a standard limit definition..

myininaya
 5 years ago
Best ResponseYou've already chosen the best response.1log base e of a right?

myininaya
 5 years ago
Best ResponseYou've already chosen the best response.1now we want to show the slope is 1 at the yintercept( or when x=0) only if a=1. f'(0)=a^0 *lna=lna lna=1 only when a=e.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0is there n easier proof because i havent learnt the general formula you used

myininaya
 5 years ago
Best ResponseYou've already chosen the best response.1it depends what you can use to prove can use that the derivative of lny =y'/y

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0yeah ive seen that before

myininaya
 5 years ago
Best ResponseYou've already chosen the best response.1y=a^x lny=lna^x lny=xlna y'/y=lna y'=ylna y'=a^x*lna

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0oh i see. i understand now thanks

myininaya
 5 years ago
Best ResponseYou've already chosen the best response.1but what him mentioned above his the fundamental definition of derivative and you have not seen it? it just means as the slopes of secants gets closer to the point of tangency we can find the slope of the tangent line

myininaya
 5 years ago
Best ResponseYou've already chosen the best response.1but you haven't seen that form there is another form f'(x0)=limx>x0[(f(x)f(x0))/(xx0)]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0no i didnt understand the last line of his one

myininaya
 5 years ago
Best ResponseYou've already chosen the best response.1the third question is that a=e^(1/A)

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0do you differentiate it then sub in a=e^(1/A) to see if it is true?

myininaya
 5 years ago
Best ResponseYou've already chosen the best response.1so we have y=Aa^x we already found that if y=a^x then y'=a^x*lna so if y=Aa^x then y'=Aa^xlna yes just put where there is an a, e^(1/A) also yintercept means x=0 so we have y'=Aa^0*lna=Alna=Aln(e^[1/A])=A*1/A*lne=1*1=1
Ask your own question
Sign UpFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.