Question

Use the definition of the derivative to compute the derivative of the given function. $$r(x)=\frac{1}{x}$$

Short answer

The derivative of \( r(x) = \frac{1}{x} \) is \( r'(x) = \frac{-1}{x^2} \).

Step-by-step solution

State the Definition of the Derivative

The derivative of a function \( f(x) \) at a point \( x \) is defined as \( f'(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h} \). In this exercise, we need to find \( r'(x) \) where \( r(x) = \frac{1}{x} \).

Setup the Difference Quotient

Substitute \( r(x) = \frac{1}{x} \) into the difference quotient formula: \( \lim_{{h \to 0}} \frac{\frac{1}{x+h} - \frac{1}{x}}{h} \). Simplify the numerator by obtaining a common denominator, which is \((x)(x+h)\):\[ \frac{1}{x+h} - \frac{1}{x} = \frac{x - (x+h)}{x(x+h)} = \frac{-h}{x(x+h)}. \]

Simplify the Expression

Replace the original difference quotient with the simplified expression from the previous step:\[ \lim_{{h \to 0}} \frac{-h}{h \, x(x+h)}. \]Cancel \( h \) in the numerator and denominator:\[ = \lim_{{h \to 0}} \frac{-1}{x(x+h)}. \]

Evaluate the Limit

Now, take the limit as \( h \to 0 \): Replace \( x+h \) with \( x \) in the limit:\[ = \frac{-1}{x \cdot x} = \frac{-1}{x^2}. \]

Conclusion: Find the Derivative

Thus, the derivative of \( r(x) = \frac{1}{x} \) is \( r'(x) = \frac{-1}{x^2} \).

Key concepts

Definition of Derivative

The derivative of a function is a fundamental concept in calculus, representing the rate at which a function changes at any given point. In more intuitive terms, it shows how the function's output changes as we make small changes to the input. This is incredibly useful in situations where you need to understand the behavior of a function, like the speed of a moving car at any given moment.
To find the derivative of a function, we use a special rule known as the "Definition of Derivative". This definition is a formula involving limits, which helps measure the instantaneous rate of change. It is given by:

• \( f'(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h} \)

This formula calculates the derivative by finding how the function changes as we take a tiny step, \( h \), toward a particular point.
For the function \( r(x) = \frac{1}{x} \), the task is to apply this definition to find its derivative.

Limit of a Function

Limits are an essential tool in calculus, allowing you to deal with quantities that approach a certain value. When you see a limit, think of it as attempting to pinpoint where a function is heading as it moves closer and closer to a particular point.
The role of limits in derivatives is crucial. In the case of a derivative, you need to evaluate the behavior of the function as the increment \( h \) approaches zero. This means you're interested in what happens to the difference quotient (the fraction given in the derivative's definition) as \( h \) becomes very tiny.
With the function \( r(x) = \frac{1}{x} \), the limit to consider is:

• \( \lim_{{h \to 0}} \frac{\frac{1}{x+h} - \frac{1}{x}}{h} \)

Successfully evaluating this limit allows us to find the derivative and understand the instantaneous rate of change of the function.

Difference Quotient

The difference quotient is a concept used to mathematically formalize the rate of change of a function. This is basically the heart of the derivative formula. The difference quotient is:

• \( \frac{f(x+h) - f(x)}{h} \)

For the function \( r(x) = \frac{1}{x} \), the difference quotient becomes:

• \( \frac{\frac{1}{x+h} - \frac{1}{x}}{h} \)

It's essentially measuring the "slope" of a secant line between two points on the curve of the function.
The point of using the difference quotient is to obtain a formula that represents the derivative by taking the limit of this quotient as \( h \) approaches zero.
This approach leads us to understand just how steep the function is at any specific point \( x \), which becomes the foundation for analyzing and predicting the behavior of more complicated functions.

Simplifying Expressions

In calculus, simplifying expressions is a key step to make problems more manageable and to clearly see what needs to be calculated. This is especially important when dealing with the derivative's difference quotient.
In the context of finding the derivative of \( r(x) = \frac{1}{x} \), we simplify by combining fractions and canceling out terms in the difference quotient:

• \( \lim_{{h \to 0}} \frac{-h}{h \cdot x(x+h)} \)

By canceling the \( h \) in the numerator and denominator, the expression simplifies to:

• \( \lim_{{h \to 0}} \frac{-1}{x(x+h)} \)

This simplification makes it much easier to evaluate the limit as \( h \) tends to zero, leading to the final outcome of the derivative.
Without simplification, limits can appear complex and confusing, making them more challenging to solve accurately.