What does the linearity of a derivative mean?
By Emma Valentine •
What does the linearity of the $n$th order derivative of a dependent variable $y$ with respect to, say, time $t$ mean?
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$\begingroup$It means that if you have two functions $f(t)$ and $g(t)$, and you want to differentiate $y(t) = f(t)+g(t)$, then it's the same as differentiating $f$, differentiating $g$, and then add the two together: $$ \frac{dy}{dt} = \frac{df}{dt} + \frac{dg}{dt} $$ This, of course, also work for $n$'th order derivatives.
There is another thing that is needed in order to call differentiation "linear", and that's the following: If your $y(t)$ is of the form $c\cdot y_0(t)$, where $c$ is a constant, then you can put the constant "outside" the differentiation: $$ \frac{dy}{dt} = c\cdot \frac{dy_0}{dt} $$
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