Derivative of the linear function

Take a look now at the following table and try to complete it:

$f (x)$	$f^{'} (x)$
$x$	$1$
$3 x$	$3$
$5 x + 2$	$5$
$10 x$	?
$8 x + 0.22$	?
$A x$	?
$A x + B$	?

Solution: $\begin{array}{ll} f (x) = 10 x & f^{'} (x) = 10 \\ f (x) = 8 x + 0.22 & f^{'} (x) = 8 \\ f (x) = A x & f^{'} (x) = A \\ f (x) = A x + B & f^{'} (x) = A \end{array}$

The type of function $f (x) = A x + B$ is called a linear function and we already learned how to find its derivative, irrespective of the constants $A$ and $B$ . As we can see, the derivative will always be $A$ .

In the first examples of the table we did not have the constant $B$ , but it does not matter since the derivative of a constant is always zero.

Note also that when $A = 0$ we are back into the derivative of a constant.