(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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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
but you haven't seen that form there is another form
no i didnt understand the last line of his one
the third question is that a=e^(1/A)
do you differentiate it then sub in a=e^(1/A) to see if it is true?
so we have
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