2.3 Q4) Hey, I need help with the following calc problem, states to use the definition below. Thank you! ### Understanding the Derivative of a FunctionIn calculus, the derivative of a function at a point provides information about the rate at which the function's value changes at that point. The formal definition of the derivative is given by the following limit:\[ f'(a) = \lim_{{x \to a}} \frac{f(x) - f(a)}{x - a} \]#### Explanation:- **\(f'(a)\)**: Represents the derivative of the function \(f(x)\) at the point \(x = a\).- **\(\lim_{{x \to a}}\)**: Indicates that we are taking the limit as \(x\) approaches \(a\).- **\(\frac{f(x) - f(a)}{x - a}\)**: - The numerator \( f(x) - f(a) \) represents the change in the function's value as \( x \) changes from \( a \) to \( x \). - The denominator \( x - a \) represents the change in the input value from \( a \) to \( x \). - Together, the fraction \( \frac{f(x) - f(a)}{x - a} \) represents the average rate of change of the function over the interval from \( a \) to \( x \). As \( x \) gets arbitrarily close to \( a \), this average rate of change approaches the instantaneous rate of change at the point \( a \), which is the value of the derivative at that point.This formula is fundamental in calculus and is used to find the slopes of tangent lines to the graph of a function, velocities in physics, and rates of change in various scientific disciplines. ### Calculating Derivatives Using the DefinitionIn this exercise, we will use the definition of the derivative to compute the derivative of the given function at a specific value.**Function and Value:**\[ f(x) = 6x^2 - x \]\[ x = 2 \]**Derivative:**\[ f'(2) = \]**Steps to Compute:**1. **Definition of the Derivative:** The derivative of a function \( f(x) \) at a point \( x = a \) is given by: \[ f'(a) = \lim_{{h \to 0}} \frac{f(a + h) - f(a)}{h} \]2. **Substitute Given Values:** Substitute \( a = 2 \) into the definition of the derivative: \[ f'(2) = \lim_{{h \to 0}} \frac{f(2 + h) - f(2)}{h} \]3. **Function Evaluation:** - \( f(2 + h) \) can be found by substituting \( 2 + h \) into the function \( f(x) \): \[ f(2 + h) = 6(2 + h)^2 - (2 + h) \] - \( f(2) \) can be computed directly from the function: \[ f(2) = 6(2)^2 - 2 \]4. **Simplify Expression:** Simplify the expression obtained and apply the limit as \( h \) approaches 0.Place your solution in the blank provided:\[ f'(2) = \boxed{} \] Be sure to simplify your calculations and provide the final numerical value.

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Answered: Use this definition to compute the…

Use this definition to compute the derivative of the function at the given value. f(x) = 6x² − x, x = 2 f'(2) =

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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