Rules of Finding Derivatives

The way to find the derivatives of basic functions will be omitted here.

Chain Rule

Most functions in use are the compositions of basic functions. The chain rule will be applied when finding their derivatives. To prove it, we only need to simply use the method of substitution, which is an important idea in mathematics.

Take ${\displaystyle f(g(x))}$ or ${\displaystyle f\circ g}$ for example:

We first let ${\displaystyle g(x)=m}$, then ${\displaystyle f^{\prime }(m)}$ means that we need to find how ${\displaystyle f(m)}$ changes while ${\displaystyle m}$ moves a little step by ${\displaystyle dm}$.

Since ${\displaystyle m=g(x)}$, ${\displaystyle m}$ differs when ${\displaystyle x}$ changes. It is natural to indicate that ${\displaystyle dm=dg=g^{\prime }(x)dx}$.

Then, for ${\displaystyle f(m)}$, we can also find that ${\displaystyle f^{\prime }(g(x))=f^{\prime }(m)dm=df}$.

In general, now we have ${\displaystyle df=f^{\prime }(g(x))dm}$ and ${\displaystyle dx=\frac{dm}{g^{\prime }(x)}}$, that is:

${\displaystyle f^{\prime }(x)=\frac{df}{dx}=f^{\prime }(g(x))g^{\prime }(x)}$, the chain rule.