Question

Finding the Derivative by the Limit Process In Exercises \(11-24,\) find the derivative of the function by the limit process. $$ f(x)=7 $$

Short answer

The derivative of the constant function \( f(x) = 7 \) is \( f'(x) = 0 \)

Step-by-step solution

Function Analysis

Identify the given function. In this exercise, the function is \( f(x) = 7 \) which is a constant function.

Understand the limit process

The limit process states that the derivative of a function at a certain point is the limit as \( h \) approaches zero of \( \frac{f(x + h) - f(x)}{h} \).

Applying the limit process

Apply the limit process to \( f(x) \). Which will give us \( f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} = \lim_{h \to 0} \frac{7 - 7}{h} = \lim_{h \to 0} \frac{ 0 }{ h } \).

Simplifying the limit

Since the numerator of the fraction is zero, the whole fraction becomes zero. Therefore, \( f'(x) = 0 \)

Key concepts

Derivative of a Constant Function

When dealing with calculus, understanding the behavior of different types of functions is crucial. A constant function is one where the output value is the same, regardless of the input. Mathematically, it's written as f(x) = c, where c is a constant.

When we seek to find the derivative, which measures the rate at which a function's output changes with respect to its input, we might guess that the derivative of a constant function is quite uneventful. And that's correct—since a constant function does not change, its rate of change is zero.

Using the limit process to find the derivative of the constant function f(x) = 7, we end up with the result f'(x) = 0. This result is universal for any constant function: the derivative will always be zero because there is no change occurring regardless of the value of x.

Limit Definition of Derivative

The limit definition of a derivative stands as one of the foundational elements of calculus. It is used to mathematically describe the precise instant rate of change of a function at a particular point. The derivative f'(x) of a function f(x) at point x is defined as:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \]
This definition hinges on the concept of a limit, which allows us to approximate the slope of the tangent line to the curve of f(x) at x. By finding the ratio of the change in the function's output to the change in input as the input's change approaches zero, the limit process pinpoints the function's local rate of change or instantaneous velocity.

This calculation embodies the derivative's primary role: to provide a way to calculate instantaneous rates of change, which is fundamental in fields such as physics, engineering, and economics. Moreover, understanding this definition helps in visualizing how a function's output value changes as its input is slightly altered, emphasizing the function's behavior around the point of interest.

Calculus Educational Content

Calculus can often seem daunting to new students; however, the goal of educational content is to break down complex concepts into digestible pieces. The focus is on offering clear explanations, practical examples, and a step-by-step approach to solving problems.

For instance, in understanding derivatives, starting with tactile examples, such as the speed of a car as it travels, can help in grasping the concept of rate of change. Illustrating the abstract with the concrete aids in building intuition, which is then reinforced with mathematical definitions and processes, like the limit definition of a derivative covered here.

Moreover, visual aids like graphs and animation can enhance learners' comprehension by showing the dynamic nature of calculus concepts. By seamlessly blending these tools and techniques, educational content in calculus ensures that students not only memorize procedures but truly understand the principles at play. Ultimately, this leads to a deeper appreciation of mathematics and its applications in the real world.