This section is aimed at students in upper secondary education in the Danish
school system, some objects will be simplified some details will be omitted.
Derivative of a Composite Function
Consider the composite function \(p(x)=f(g(x))\), its derivative is
\(p'(x)=f'(g(x))g'(x)\).
Proof.
Lets start with the secant slope
\begin{align}
\frac{p(x+h)-p(x)}{h}=&\frac{f(g(x+h))-f(g(x))}{h}\\
=&\frac{f(g(x+h))-f(g(x))}{h}\frac{g(x+h)-g(x)}{g(x+h)-g(x)}\\
=&\frac{f(g(x)+g(x+h)-g(x))-f(g(x))}{g(x+h)-g(x)}\frac{g(x+h)-g(x)}{h}\\
=&\frac{f(g(x)+k)}{k}\frac{g(x+h)-g(x)}{h}\to f'(g(x))g'(x)
\end{align}
where \(k=g(x+h)-g(x)\to0\) for \(h\to0\) since \(g\) is continuous.
Derivative of the Natural Logarithm
An immediate consequence of this formula is that the derivative of the
natural logarithm is \(1/x\).
Proof.
Consider that \(p(x)=x=e^{\ln(x)}\) where \(f(x)=e^x\) and \(g(x)=
\ln(x)\) then we get that
\begin{align}
1=p'(x)=&f'((g(x))g'(x)\\
=&e^{\ln(x)}g'(x)\\
=&xg'(x)\implies g'(x)=\frac{1}{x}
\end{align}