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.

24. \(f\left( x \right) = 4 + 8x - 5{x^2}\)

Short answer

The derivative of the function is \(8 - 10x\). The domain of the function \(f\) and derivative \(f'\) is \(\mathbb{R}\).

Step-by-step solution

Condition for derivative of the function

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:

\(\begin{aligned}f'\left( x \right) & = \mathop {\lim }\limits_{h \to 0} \frac{{f\left( {x + h} \right) - f\left( x \right)}}{h}\\ & = \mathop {\lim }\limits_{h \to 0} \frac{{\left( {4 + 8\left( {x + h} \right) - 5{{\left( {x + h} \right)}^2}} \right) - \left( {4 + 8x - 5{x^2}} \right)}}{h}\\ & = \mathop {\lim }\limits_{h \to 0} \frac{{4 + 8x + 8h - 5\left( {{x^2} + 2xh + {h^2}} \right) - 4 - 8x + 5{x^2}}}{h}\\ & = \mathop {\lim }\limits_{h \to 0} \frac{{8h - 5{x^2} - 10xh - 5{h^2} + 5{x^2}}}{h}\\ & = \mathop {\lim }\limits_{h \to 0} \frac{{8h - 10xh - 5{h^2}}}{h}\\ & = \mathop {\lim }\limits_{h \to 0} \frac{{h\left( {8 - 10x - 5h} \right)}}{h}\\ & = \mathop {\lim }\limits_{h \to 0} \left( {8 - 10x - 5h} \right)\\ & = 8 - 10x\end{aligned}\)

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

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

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