Hey! I was recently trying to prove the derivative of a^x.

After trying it myself I ended up having to Google it because I don't seem to get it quite right. Here is what I tried to do:

\(y = {a^{x}}\)

\(ln(y) = x*ln(a)\)

\(\frac{d}{dx} ln(y) = \frac{d}{dx} x*ln(a)\)

Since ln(a) is a constant I just take the drivative of x which is one.

\(\frac{1}{y} = ln(a) \)

From above we see that y = a^x so I substitute it in there.

\(\frac{1}{a^{x}}= ln(a) \)

Rearranging gives me:

\(1= ln(a) * a^{x}\)

Here comes my problem, the result is supposed to be:

\(\frac{dy}{dx}= ln(a)*a^{x}\)

the dy/dx is supposed to come up on the left hand side as I take the derivative of ln(y) and get 1/y. I don't understand why this happens and I haven't seen any explanation behind for that specific step. Everything else is logical, just logarithmic/exponential properties. I would appreciate any help, thanks in advance!

Quazars
Nov 9, 2015

#1**+15 **

Best Answer

y = a^x take the ln of both sides

lny = lna^x and we can write

lny = ln a^x exponentiate both sides

e ^(ln y) = e^(ln a^x)

y = e^(ln a^x)

y = e^(x ln a) take the derivative

y ' = lna * e^(x ln a)

y ' = lna * e^(ln a^x)

y ' = lna * a^x and we can write

dy / dx = (ln a) * a^x

CPhill
Nov 9, 2015

#3**+5 **

At the point where you have \(\frac{d}{dx}\ln(y),\)

you are required to differentiate a function of y wrt x.

To do this you should be using the function of a function rule, which, in this case, takes the form

\(\displaystyle \frac{df(y)}{dx}=\frac{df(y)}{dy}\times\frac{dy}{dx}.\)

Guest Nov 10, 2015