proof of derivative of a^x

how do you prove that the derivative of a^x = (a^x)(ln a)?

i tried using logarithmic differentiation:
y = a^x
ln y = ln (a^x)
ln y = x (ln a)
d/dx (ln y) = d/dx [x (ln a)]
(1/y) (dy/dx) = (1)(ln a) + (x)(1/a)
dy/dx = y [(ln a) + (x/a)]
so if i substitute
dy/dx = a^x [(ln a) + (x/a)]

did i do something wrong? how come it's not coming out correctly? i used logarithmic differentiation to prove that the derivative of e^x = e^x and it worked. how come it doesn't work for proving this?

Quote:

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Originally Posted by oblixps

y = a^x
ln y = ln (a^x)
ln y = x (ln a)
d/dx (ln y) = d/dx [x (ln a)]
(1/y) (dy/dx) = (1)(ln a) + (x)(1/a)
dy/dx = y [(ln a) + (x/a)]
so if i substitute
dy/dx = a^x [(ln a) + (x/a)]
?

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You're differentiating with respect to . So, is just simply a constant which you can pull out, i.e.:

oh wow i can't believe i didn't see that...
thanks!!

i tried using logarithmic differentiation:
y = a^x
ln y = ln (a^x)
ln y = x (ln a)
d/dx (ln y) = d/dx [x (ln a)]
(1/y) (dy/dx) = (1)(ln a) + (x)(1/a)
dy/dx = y [(ln a) + (x/a)]
so if i substitute
dy/dx = a^x [(ln a) + (x/a)]

did i do something wrong? how come it's not coming out correctly?
YES you did.
Your technique is correct.
BUT a is a constant, so there's no product rule here

Now continue...