Answer
$4^{-x}-4^{-x}\ln 4 (x+1)$
Work Step by Step
We have: $f(x)=4^{-x} (x+1)$
We differentiate both sides with respect to $x$.
$f^{\prime}(x)=\dfrac{d}{dx} [4^{-x} (x+1)] \\=4^{-x} +(x+1) \times (4^{-x} \ln 4) \times (-1)$
Simplify to obtain:
$f^{\prime}(x)=4^{-x}-4^{-x}\ln 4 (x+1)$
