Differentials and integrals: Differentials
Calculation rules for differentials
The aforementioned relationship between differentials and derivatives makes the following translation of mathematical rules to differentiate rules for calculating differentials possible:
Calculation rules for differentials \[\begin{aligned} \dd\bigl(c\cdot f(t)\bigr) &= c\cdot \dd\bigl(f(t)\bigr), \text{ for any constant }c\\ \\ \dd\bigl(f(t)+g(t)\bigr) &= \dd\bigl(f(t)\bigr)+\dd\bigl(g(t)\bigr)\;\;\; (\text{sum rule})\\ \\ \dd\bigl(f(t)\cdot g(t)\bigr) &= g(t)\cdot \dd\bigl(f(t)\bigr)+f(t)\cdot \dd\bigl(g(t)\bigr)\;\;\; (\text{product rule})\\ \\ \dd\left(\frac{f(t)}{g(t)}\right) &= \frac{g(t)\cdot \dd\bigl(f(t)\bigr)-f(t)\cdot \dd\bigl(g(t)\bigr)}{\bigl(g(t)\bigr)^2}\;\;\; (\text{quotient rule})\\ \\ \dd\Bigl(f\bigl(g(t)\bigr)\Bigr) &= f'\bigl(g(t)\bigr)\cdot \dd\bigl(g(t)\bigr)= f'\bigl(g(t)\bigr)\cdot g'(t)\,\dd t\;\;\; (\text{chain rule})\\ \end{aligned}\]
In a short and more clear notation, the first three calculation rules can be written as:
Shorthand notation of rules for differentials \[\begin{aligned} \dd(c\cdot f) &= c\cdot \dd f, \text{ for any constant }c\\ \\ \dd(f+g) &= \dd f+\dd g\;\;\; (\text{sum rule})\\ \\ \dd(f\cdot g) &= g\cdot \dd f+f\cdot \dd g\;\;\; (\text{product rule})\\ \\ \dd\left(\frac{f}{g}\right) &= \frac{g\cdot \dd f-f\cdot \dd g}{g^2}\;\;\; (\text{quotient rule})\\ \end{aligned}\]
We give some examples of working with the calculation rules for differentials.
Via the constant factor rule and the sum rule for differentials, and via the derivative of a power function, this goes as follows: \[\begin{aligned} \dd(8t^2+7t-2) &= \dd(8t^2)+\dd(7t)+\dd(-2)\\ &\phantom{uvwxyz}\blue{\text{sum rule}}\\ &= 8\,\dd(t^2)+7\,\dd t+\dd(-2)\\ &\phantom{uvwxyz}\blue{\text{constant factor rule}}\\ &=8\cdot 2t\,\dd t+7\,\dd t+0\\ &\phantom{uvwxyz}\blue{\text{derivative of a power}}\\ &= 16t\,\dd t+7\,\dd t\\\ &\phantom{uvwxyz}\blue{\text{simplification}}\\ &= (16t+7)\,\dd t\\ &\phantom{uvwxyz}\blue{\text{rewriting}}\end{aligned}\]
You can also do the exercise by reading \(8t^2+7t-2\) as a function \(f(t)\) and calculating the derivative \(f'(t)\). After all, \(\dd\bigl(f(t)\bigr) = f'(t)\,\dd t\).
