What is the derivative of $4^x$ ?

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Answer 1 · 2015-08-24T16:13:30

$d/dx(4^x) = 4^x ln4$

In general for $b>0$ , we have
$d/dx(b^x) = b^x lnb$

And when we need the chain rule, we have

$d/dx(b^u) = b^u (lnb) d/dx(u)$

Proof of General rule

$b^x = (e^lnb)^x$ $" "$ (remember that $e^lnu = u$ )

So
$b^x = e^(xlnb)$ $" "$ (because $(a^r)^s = a^(rs) = a^(sr)$ )

$d/dx(e^(xlnb))$ can be found by the chain rule:

$d/dx(e^(xlnb)) = e^(xlnb) d/dx(xlnb)$

$= e^(xlnb) lnb$ $" "$ ( $lnb$ is a constant)

$= b^x lnb$ $" "$ (reverse the first two steps.) $square$

What is the derivative of 4^x4^x?