Proof of derivative a^x

1. Apr 13, 2012

Nano-Passion

I get a wrong expression when I try to take to take the derivative of (ln a) x

1. The problem statement, all variables and given/known data
Prove $$\frac{d}{dx} a^x = (ln a)a^x$$

2. Relevant equations
By definition, $$a^x = e^{e(ln a)x}$$

3. The attempt at a solution
$$\frac{d}{dx} a^x = (ln a)a^x$$
$$\frac{d}{dx}e^u\frac{du}{dx}$$
Solving for du/dx gives $$\frac{d}{dx}(ln a)x$$
Let $$u = ln a . v = x$$
From the product rule,
$$u'v+v'u$$ which gives
$$\frac{du}{dx} = \frac{x}{a} + ln a$$
so plugging it back to original expression gives
$$\frac{d}{dx}a^x = e^{e(ln a)x} * (\frac{x}{a} + ln a)$$

Which is obviously wrong.

Last edited: Apr 13, 2012
2. Apr 13, 2012

lol_nl

This should be $$a^x = (e^{ln(a)})^{x} = e^{(ln(a)) x}.$$ Now taking the derivate with respect to x should bring you directly to your answer (remember that ln(a) is a constant).

3. Apr 13, 2012

Nano-Passion

Whoops, a is a constant.. it completely skipped my mind. Thank you!!