How to correctly take the derivative of a^x?

[Nano-Passion]

[Answered] Proof of derivative a^x

[Answered] Thanks.

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

Homework Statement

Prove \frac{d}{dx} a^x = (ln a)a^x

Homework Equations

By definition, a^x = e^{e(ln a)x}

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.

 

[lol_nl]

Homework Equations

By definition, a^x = e^{e(ln a)x}

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).

 

[Nano-Passion]

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).

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