## Section7.5Rules for Differentials

We can eliminate $y$ from (7.4.5) by writing

$$d(x^2) = 2x\,dx ,\label{zapex1}\tag{7.5.1}$$

and we can go even further by replacing the ubiquitous variable $x$ by any physical or geometric quantity, such as $u\text{.}$ The beauty of this approach is that differentiation is easy once you have convinced yourself of a few basic rules.

Let's start with some simple functions. For instance, the power rule for derivatives says that

$$\frac{d}{du}(u^n) = nu^{n-1}\label{dpower}\tag{7.5.2}$$

which in differential notation becomes $d(u^n)=nu^{n-1}\,du\text{.}$

##### Notation.

We use $u$ and later $v$ to denote any quantity. It might be the case that $u=x\text{,}$ or that $u=f(x)\text{,}$ or that $u=f(x,y)\text{,}$ or that $u$ depends on other quantities. it doesn't matter.

##### Usage.

“Taking the differential” or “zapping with $d$” is an operation; $d$ itself is an operator, that acts on functions.

Applying this construction to the derivatives of elementary functions, we obtain the basic differentiation formulas in differential form, namely:

\begin{align*} d\left(u^n\right) \amp = nu^{n-1} \,du ,\\ d\left(e^u\right) \amp = e^u \,du ,\\ d(\sin u) \amp = \cos u \,du ,\\ d(\cos u) \amp = -\sin u \,du ,\\ d(\ln u) \amp = \frac{1}{u} \> du ,\\ d(\tan u) \amp = \frac{1}{\cos^2u} \> du . \end{align*}

So how do we use these formulas to compute derivatives?