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I am trying to compute simple derivatives of simple functions, but I got stuck on $\frac{d}{dx}a^x=(\ln{a})a^x$.

I suppose the proof is a simple corollary of $\frac{d}{dx}e^x=e^x$, but I am unable to find it. Can anybody help me?

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up vote 2 down vote accepted

Hint $a^x=e^{\ln (a^x)} = e^{x \ln a}$

You can also use logarithmic differentiation...

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Thanks! I had applied the chain rule, but then I was making a (rather silly) mistake... – Danilo Piazzalunga Dec 27 '11 at 22:16

Write $a^x$ as: $a^x=e^{\ln a^x}=e^{x\ln a}$ and use the chain rule.

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Chain rule: $2^x$ is the same as $e^{(\log_e 2)x}$. To differentiate $(\log_e 2)x$, remember that $\log_e 2$ is a constant.

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