Differentiate exponentials and logarithms with their inner factors

$d(4e^{x^2+1})/dx=4e^{x^2+1}(2x)=8xe^{x^2+1}$, for every real $x$.

$d(3^{\tan x})/dx=3^{\tan x}\ln3\,\sec^2x$, where $\cos x\ne0$. The Teacher completion retains both the base factor $\ln3$ and derivative of the inner tangent.

$d[\ln(x+x^3)]/dx=(1+3x^2)/(x+x^3)$. Since $x+x^3=x(1+x^2)$, the real logarithm requires $x>0$. A derivative expression defined elsewhere does not extend the logarithm’s domain.

Write $\log_3(x^2+4)=\ln(x^2+4)/\ln3$. Then the derivative is $2x/[(\ln3)(x^2+4)]$. The Teacher’s filled change-of-base and chain steps are retained. Since $x^2+4>0$, the formula is valid for all real $x$.