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.

21. \(f\left( x \right) = 3x - 8\)

Short answer

The derivative of the function is 3. 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:

\(\begin{aligned}f'\left( x \right) & = \mathop {\lim }\limits_{h \to 0} \frac{{f\left( {x + h} \right) - f\left( x \right)}}{h}\\ & = \mathop {\lim }\limits_{h \to 0} \frac{{\left( {3\left( {x + h} \right) - 8} \right) - \left( {3x - 8} \right)}}{h}\\ & = \mathop {\lim }\limits_{h \to 0} \frac{{3x + 3h - 8 - 3x + 8}}{h}\\ & = \mathop {\lim }\limits_{h \to 0} \frac{{3h}}{h}\\ & = \mathop {\lim }\limits_{h \to 0} 3\\ & = 3\end{aligned}\)

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 \(\mathbb{R}\).