Question

Give the function f(x) = a^x for some real constant a ?

Answer 1

i) Letting `y = a^x`, we see that

`lny = ln(a^x)`

`lny = xlna`

Since `a` is a constant, we can say

`1/y(dy/dx) = lna`

`dy/dx = ylna`

`dy/dx= a^xlna`

Therefore, `M(a) = lna`.

ii) Recall the definition of the derivative is `f'(x) = lim_(h-> 0) (f(x+ h) -f (x))/h`.

`f'(x) = lim_(h-> 0) (a^(x + h) - a^x)/h`

`f'(x) = lim_(h->0) (a^xa^h - a^x)/h`

`f'(x) = lim_(h->0) (a^x(a^h - 1))/h`

`f'(x) = a^x lim_(h-> 0) (a^h - 1)/h`

iii) We seek to prove that `lim_(h->0) (a^h - 1)/h = lna`

Since we're of the form `0/0` when we attempt direct substitution, we may apply l'hospitals rule to evaluate the limit. We derived the derivative of `a^h` in part `(i)` as being `a^hln(a)`.

`L = lim_(h->0) (a^hlna)/1`

`L = (a^0lna)/1`

`L = lna`

As required.

Hopefully this helps!