Quick questions on Derivatives of exponential and logarithmic functions (QCE Mathematical Methods Unit 3)

The exponential function with base $e$ is its own derivative.

For $a > 0$ and $a \neq 1$, use the change of base $a^x = e^{x \ln a}$, which gives

Before differentiating a complicated logarithm, simplify using logarithm laws. For example,

The reason $e^{kt}$ models growth and decay is that its derivative is proportional to itself: if $Q(t) = Q_0 e^{kt}$ then $Q'(t) = kQ_0 e^{kt} = kQ$. So the rate of change at any instant is exactly $k$ times the current amount, which is the defining feature of exponential change. A positive $k$ gives growth (the rate rises as the quantity rises) and a negative $k$ gives decay (the rate of loss shrinks as the quantity shrinks). Recognising $Q' = kQ$ as the signature of an exponential model is a recurring QCAA theme and connects this dot point to differential equations.