anonymous
  • anonymous
(another characterisation of y = e^x -- it is the only exponential function whose gradient at its y-intercept 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)
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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anonymous
  • anonymous
f(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
  • anonymous
im alittle confused
anonymous
  • anonymous
why?

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anonymous
  • anonymous
is that a general formula for intergrating exponentials?
anonymous
  • anonymous
no...
anonymous
  • anonymous
@and: differentiating.
anonymous
  • anonymous
its the fundamental process for evaluating derivatives..
anonymous
  • anonymous
oh sorry yeah i ment diffrentiating
anonymous
  • anonymous
the part where ive replaced (a^h - 1 )/ h as loga where h tends to zero is a standard limit definition..
myininaya
  • myininaya
log base e of a right?
anonymous
  • anonymous
yeah
myininaya
  • myininaya
or natural log of a?
myininaya
  • myininaya
cool same page then
anonymous
  • anonymous
ln a
myininaya
  • myininaya
now we want to show the slope is 1 at the y-intercept( or when x=0) only if a=1. f'(0)=a^0 *lna=lna lna=1 only when a=e.
myininaya
  • myininaya
only if a=e
myininaya
  • myininaya
*
anonymous
  • anonymous
is there n easier proof because i havent learnt the general formula you used
myininaya
  • myininaya
it depends what you can use to prove can use that the derivative of lny =y'/y
anonymous
  • anonymous
yeah ive seen that before
myininaya
  • myininaya
y=a^x lny=lna^x lny=xlna y'/y=lna y'=ylna y'=a^x*lna
anonymous
  • anonymous
oh i see. i understand now thanks
myininaya
  • myininaya
but 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
  • myininaya
but you haven't seen that form there is another form f'(x0)=limx->x0[(f(x)-f(x0))/(x-x0)]
anonymous
  • anonymous
no i didnt understand the last line of his one
myininaya
  • myininaya
the third question is that a=e^(1/A)
anonymous
  • anonymous
do you differentiate it then sub in a=e^(1/A) to see if it is true?
myininaya
  • myininaya
so 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 y-intercept means x=0 so we have y'=Aa^0*lna=Alna=Aln(e^[1/A])=A*1/A*lne=1*1=1
anonymous
  • anonymous
ok thanks heaps :)

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