### Chain Rule & Integration by Substitution

We already know that the exponential function $e^x$ is its own tangent/gradient function. However, there are other exponential functions such as $2^x$, $3^x$ and so on.

Are their tangent/gradient functions the same or does something else happen?

What can we say about $1^x$ or $0.5^x$, for example?

Here is a series of structured questions leading to finding the derivative of $a^x$ with respect to $x$ for a positive number $a$.

If $e^k=2$, what is $k$?

Which is steeper, $e^x$ or $2^x$?

Can we write $2^x$ in the form $e^{kx}$, where $k$ is a real number?

What is the derivative of $e^{kx}$ with respect to $x$, where $k$ is a real number?

How can we differentiate $2^x$ with respect to $x$?

Given the above questions, how would we differentiate $a^x$, where $a$ is a positive real number?