# What is the derivative of 3^x ?

What is the derivative of ${3}^{x}$ ?
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abonirali59
Begin by letting $y={3}^{x}$
now take the ln of both sides.
$\mathrm{ln}y={\mathrm{ln}3}^{x}⇒\mathrm{ln}y=x\mathrm{ln}3$
differentiate implicitly with respect to x
$⇒\frac{1}{y}\frac{dy}{dx}=\mathrm{ln}3$
Now $y={3}^{x}⇒\frac{dy}{dx}={3}^{x}\mathrm{ln}3$
This result can be generalised as follows
$\frac{d}{dx}\left({a}^{x}\right)={a}^{x}\mathrm{ln}a$

psor32
Explanation:
$\frac{d}{dx}\left({3}^{x}\right)$
$=\frac{d}{dx}\left({e}^{x\mathrm{ln}\left(3\right)}$ (since ${a}^{b}={e}^{b\mathrm{ln}\left(a\right)}$)
$=\frac{d}{dx}\left(x\mathrm{ln}\left(3\right)\right)\frac{d}{d\left(x\mathrm{ln}\left(3\right)\right)}\left({e}^{x\mathrm{ln}\left(3\right)}\right)$
$=\mathrm{ln}\left(3\right){e}^{x\mathrm{ln}\left(3\right)}$
$=\mathrm{ln}\left(3\right){3}^{x}$