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When finding derivatives, a useful tool is $(a^x)' = (a^x)ln(a)$. Now say I have $2^{3x}$. When trying to find the derivative, are these two equivalent? $(2^x)ln(2)(3)$ and $(3x)2^{3x - 1}$

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They are not equivalent, the first derivative is given by $$(2^{3x})'=2^{3x}\ln(2)\cdot 3$$ by the chain rule.

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  • $\begingroup$ Why doesn't the power rule apply here as well? $\endgroup$ – ming Dec 10 '18 at 17:25
  • $\begingroup$ Since $$(a^x)'=a^x\ln(a)$$ and your case is $$a=2$$ $\endgroup$ – Dr. Sonnhard Graubner Dec 10 '18 at 17:29
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Both $(2^x)ln(2)(3)$ and $(3x)2^{3x - 1}$ are wrong !

We have $2^{3x}=8^x$, hence $(2^{3x})'=8^x \ln (8).$

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  • $\begingroup$ So was the rule taught to me incorrect? $( a x ) ′ =( a x )ln(a)$ $\endgroup$ – ming Dec 10 '18 at 17:58

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