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anonymous

  • 5 years ago

(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)

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  1. watchmath
    • 5 years ago
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    I 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*}

  2. anonymous
    • 5 years ago
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    but i need part b and c aswell lol

  3. watchmath
    • 5 years ago
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    well for part b) what is the x-part of the y-intercept? then plug in that x to the result in part a).

  4. anonymous
    • 5 years ago
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    how do you prove that the gradient is 1

  5. watchmath
    • 5 years ago
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    you need to show that y'=1 :) (at the y-intercept)

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