Question

(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)

Answer 1

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*}

Answer 2

but i need part b and c aswell lol

Answer 3

well for part b) what is the x-part of the y-intercept? then plug in that x to the result in part a).

Answer 4

how do you prove that the gradient is 1

Answer 5

you need to show that y'=1 :) (at the y-intercept)