Calculus: Early Transcendentals, 8th Edition
Authors: James Stewart
ISBN-13: 978-1285741550
See our solution for Question 21E from Chapter 2.8 from Stewart's Calculus, 8th Edition.
Problem 21E
Chapter:
Problem:
Step-by-Step Solution
Given information
We have to find the following function\[f\left( x \right) = 3x - 8\]We have to find the derivative of the function using definition of derivative. The formula is: \[f'\left( x \right) = \mathop {\lim }\limits_{h \to 0} \dfrac{{f\left( {x + h} \right) - f\left( x \right)}}{h}\]
Step 1: The derivative
\[\begin{array}{l}f'\left( x \right) = \mathop {\lim }\limits_{h \to 0} \dfrac{{f\left( {x + h} \right) - f\left( x \right)}}{h}\\ = \mathop {\lim }\limits_{h \to 0} \dfrac{{\left[ {3\left( {x + h} \right) - 8} \right] - \left[ {3x - 8} \right]}}{h}\\ = \mathop {\lim }\limits_{h \to 0} \dfrac{{\left[ {3x + 3h - 8 - 3x + 8} \right]}}{h}\\ = \mathop {\lim }\limits_{h \to 0} \dfrac{{3h}}{h}\\ = \mathop {\lim }\limits_{h \to 0} 3\\ = 3\end{array}\]Therefore, The derivative is \[f'\left( x \right) = 3\]
Step 2: Domain of the function
Domain is given by all possible values of x for which the function exists. The given function is a linear function, hence for all real values of x, the function exists. Therefore, Domain of function is \[{D_f} = \left( { - \infty ,\infty } \right)\]
Step 3: Domain of the Derivative
The derivative function is a constant function, hence for all real values of x, the function exists and is equal to a constant.. Therefore, Domain of function is \[{D_{f'}} = \left( { - \infty ,\infty } \right)\]
We have to find the following function\[f\left( x \right) = 3x - 8\]We have to find the derivative of the function using definition of derivative. The formula is: \[f'\left( x \right) = \mathop {\lim }\limits_{h \to 0} \dfrac{{f\left( {x + h} \right) - f\left( x \right)}}{h}\]
Step 1: The derivative
\[\begin{array}{l}f'\left( x \right) = \mathop {\lim }\limits_{h \to 0} \dfrac{{f\left( {x + h} \right) - f\left( x \right)}}{h}\\ = \mathop {\lim }\limits_{h \to 0} \dfrac{{\left[ {3\left( {x + h} \right) - 8} \right] - \left[ {3x - 8} \right]}}{h}\\ = \mathop {\lim }\limits_{h \to 0} \dfrac{{\left[ {3x + 3h - 8 - 3x + 8} \right]}}{h}\\ = \mathop {\lim }\limits_{h \to 0} \dfrac{{3h}}{h}\\ = \mathop {\lim }\limits_{h \to 0} 3\\ = 3\end{array}\]Therefore, The derivative is \[f'\left( x \right) = 3\]
Step 2: Domain of the function
Domain is given by all possible values of x for which the function exists. The given function is a linear function, hence for all real values of x, the function exists. Therefore, Domain of function is \[{D_f} = \left( { - \infty ,\infty } \right)\]
Step 3: Domain of the Derivative
The derivative function is a constant function, hence for all real values of x, the function exists and is equal to a constant.. Therefore, Domain of function is \[{D_{f'}} = \left( { - \infty ,\infty } \right)\]