Hi, this was on my last calc test, and I wanted to make sure I understood why I got it wrong.

Find the derivative of

(22^x)

ln(2)22^x

My guess now that I realize it's wrong:

ln(4)(4x)

Thanks!

$2^{1+2x}\text{ln}(2)$

Hello, rocapp!

This requires some fancy footwork . . .

$\text{Find the derivative of: }\:y \;=\;2^{2^x}$

$\text{Take logs: }\:\ln(y) \:=\:\ln\left(2^{2^x}\right) \quad\Rightarrow\quad \ln{y}\;=\;2^x\!\cdot\!\ln 2$

$\text{Differentiate implicitly: }\:\frac{1}{y}\!\cdot\!y' \;=\;2^x\!\cdot\!(\ln 2)^2$

$\text{Therefore: }\:y' \;=\;y\!\cdot\!2^x\!\cdot\!(\ln2)^2 \quad\Rightarrow\quad y'\;=\;2^{2^x}\!\cdot2^x\!\cdot\!(\ln2)^2$