Derivative of the linear function

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

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

The type of function $f (x) =Ax+B$ 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.