I've the text below given in my notes:
Derivative of linear function: Let $R:X\to Y$ be a linear function .Then $R':X\to L(X,Y)$ is a constant function with the constant value $R\in L(X,Y)$ i.e. $R'(a)=R$ for all a $\in X$.That is , $$R'(a)=R$$ for all a,x $\in X$.
Can anyone explain the above definition with help of an example clearly stating what it means?
$\endgroup$2 Answers
$\begingroup$By definition, the derivative (if exists) of $f$ in $x_0\in X$ is a linear function $$f'(x_0):X\longrightarrow Y$$ s.t. $$f(x_0+h)=f(x_0)+f'(x_0)(h)+o(h).$$ When $f=R$ linear, by this linearity $$R(x_0+h)=R(x_0)+R(h)$$ and $$f(x_0)=R(x_0),$$ $$f'(x_0)(h)=R(h),$$ $$o(h)=0.$$
Example: $$R(x_1,x_2)=\pmatrix{1&2\cr3&4}\pmatrix{x_1\cr x_2}=\cdots$$ and the matrix of $R'$ is given by the partial derivatives of $R$...
$\endgroup$ 8 $\begingroup$Let $f(x)$ be a linear function of $x$.
In general, the function $f$ then takes the form
$$f(x) = mx + b$$
where $m$ is the slope and $b$ is the y-intercept of the graph of $f$, which is nothing but a non-vertical line.
Differentiating:
$$f'(x) = m$$
which gives the derivative $f'$ as the slope of the line
$$y = f(x) = mx + b.$$
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