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Section 6.8 Differentials: Summary

We summarize here both the basic differentiation formulas from Section 6.5 and the fundamental rules from Section 6.6, in differential form. We have:

\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 ,\\ d(u+cv) \amp = du + c \,dv ,\\ d(uv) \amp = u \,dv + v \,du ,\\ d\left(\frac{u}{v}\right) \amp = \frac{v \,du - u \,dv}{v^2} , \end{align*}

where \(c\) is a constant. Each of these rules can be reinterpreted as a derivative rule by deviding both sides by \(du\text{,}\) and as an integration rule by integrating both sides (and using the Fundamental Theorem of Calculus in the form \(\int df=f\)).

We have added the quotient rule at the end of this list, which can be obtained quickly from the power and product rules by writing \(\frac{u}{v}=u\frac{1}{v}\text{.}\) (Similarly, any one rule for differentiating a trigonometric function can be used to derive the others, and, as we saw in Section 6.6, the rule for the exponential function can be used to derive the rule for logarithms – or vice versa. Nonetheless, each of these rules is used often enough that it is helpful to include them all.)

We reiterate that there is no need to add the chain rule to this list, nor the rules for inverse functions or implicit differentiation.

Finally, a nice mnemonic to help you avoid mixing up differentials with derivatives, especially at first, is to think of terms involving \(d\) as small. Derivatives are the ratios of small quantities, but they are not themselves small. And each term in an equation must have the same character, big or small. Put differently, the \(d\)s must balance; (infinitesimally) small quantities can never be equal to (finite) big quantities.