Question

Find the derivative of the function using the definition of the derivative. State the domain of the function and the domain of the derivative.

22. \(f\left( x \right) = mx + b\)

Short answer

The derivative of the function is \(m\). The domain of the function \(f\) and derivative \(f'\) is \(\mathbb{R}\).

Step-by-step solution

Condition for derivative

The derivative of the function \(f\) at any number \(x\):

\(f'\left( x \right) = \mathop {\lim }\limits_{h \to 0} \frac{{f\left( {x + h} \right) - f\left( x \right)}}{h}\)

The function \(f'\) is called the derivative of the function.

Determine the derivative of the function and domain of a function

Evaluate the derivative of the function as shown below:

\(\begin{aligned}f'\left( x \right) & = \mathop {\lim }\limits_{h \to 0} \frac{{f\left( {x + h} \right) - f\left( x \right)}}{h}\\ & = \mathop {\lim }\limits_{h \to 0} \frac{{\left( {m\left( {x + h} \right) + b} \right) - \left( {mx + b} \right)}}{h}\\ & = \mathop {\lim }\limits_{h \to 0} \frac{{mx + mh + b - mx - b}}{h}\\ & = \mathop {\lim }\limits_{h \to 0} \frac{{mh}}{h}\\ & = \mathop {\lim }\limits_{h \to 0} m\\ & = m\end{aligned}\)

The domain of the function \(f\) is the set of all real numbers.

The domain of the derivative \(f'\) is \(\mathbb{R}\).

Thus, the domain of the function \(f\) and derivative \(f'\) is \(\mathbb{R}\).