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!