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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?

coool
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2 Answers2

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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 $R$...

Martín-Blas Pérez Pinilla
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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

$$y = f(x) = mx + b.$$

Jose Arnaldo Bebita Dris
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  • please help explaining how does it show that derivative is a constant function... – coool Nov 04 '14 at 08:00
  • @coool, the slope of any non-vertical line is a real number (i.e., a constant). Therefore, the derivative of a linear function, which is nothing else but the slope of the graph of the function, is constant. – Jose Arnaldo Bebita Dris Nov 04 '14 at 08:02
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    thanks for explaining in $1$-D but the definition involves case having more than one-variable.. – coool Nov 04 '14 at 08:03