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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)

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watchmath
 5 years ago
Best ResponseYou've already chosen the best response.0I am just in the mood of doing part (a) \begin{align*} y&=a^x\\ \ln y&=x\ln a\\ \frac{1}{y}y'&=\ln a\\ y'&=y\ln a\\ y'&=a^x\ln a \end{align*}

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0but i need part b and c aswell lol

watchmath
 5 years ago
Best ResponseYou've already chosen the best response.0well for part b) what is the xpart of the yintercept? then plug in that x to the result in part a).

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0how do you prove that the gradient is 1

watchmath
 5 years ago
Best ResponseYou've already chosen the best response.0you need to show that y'=1 :) (at the yintercept)
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