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.