Question

Use the definition of the derivative to compute the derivative of the given function. $$r(s)=\frac{1}{s-2}$$

Short answer

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

Step-by-step solution

State the Definition of the Derivative

The derivative of a function, denoted as \( f'(x) \), is defined as the limit: \[ f'(x) = \lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h} \]. In our case, the function is \( r(s) = \frac{1}{s-2} \).

Apply the Definition to the Function

Substitute \( r(s) \) into the definition: \[ r'(s) = \lim_{{h \to 0}} \frac{{\frac{1}{s+h-2} - \frac{1}{s-2}}}{h} \]. This setup prepares us to evaluate the limit.

Simplify the Numerator

To simplify \( \frac{1}{s+h-2} - \frac{1}{s-2} \), find a common denominator: \( (s+h-2)(s-2) \). The expression becomes: \[ \frac{(s-2) - (s+h-2)}{(s+h-2)(s-2)} = \frac{-h}{(s+h-2)(s-2)} \].

Substitute Back into the Derivative Definition

Now the expression for the derivative becomes: \[ r'(s) = \lim_{{h \to 0}} \frac{\frac{-h}{(s+h-2)(s-2)}}{h} = \lim_{{h \to 0}} \frac{-h}{h(s+h-2)(s-2)} \].

Simplify the Expression

Cancel the \( h \) terms: \[ r'(s) = \lim_{{h \to 0}} \frac{-1}{(s+h-2)(s-2)} \].

Evaluate the Limit

Substitute \( h = 0 \) into the expression: \[ r'(s) = \frac{-1}{(s-2)^2} \]. Thus, the derivative of \( r(s) = \frac{1}{s-2} \) is \( r'(s) = \frac{-1}{(s-2)^2} \).

Key concepts

Definition of the Derivative

In calculus, the derivative is a central concept that describes how a function changes as its input changes. Simply put, it tells you the rate at which a function is changing with respect to one of its variables. The formal definition of the derivative of a function \( f(x) \), denoted as \( f'(x) \), is given by the limit:\[f'(x) = \lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h}\]This expression represents the slope of the tangent line to the function at any point \( x \). It's like zooming in super close to a curve until it almost looks like a straight line.

• \( f(x+h) \) is the function with a slight increase in \( x \)
• \( f(x) \) is the current value of the function
• \( h \) is the small increment approaching zero

Understanding this formula helps us calculate how functions react to changes in their input values.

Limit Process

The limit process is an essential tool in calculus, allowing us to capture the behavior of functions as inputs approach a particular value. Understanding this behavior is crucial for computing derivatives. In the definition of the derivative, the limit \( \lim_{{h \to 0}} \) is used to express how the difference quotient approaches a value when the increment \( h \) becomes infinitesimally small.The difference quotient \( \frac{{f(x+h) - f(x)}}{h} \) calculates the average rate of change over a small interval \( h \).

• As \( h \) approaches zero, the average rate of change turns into the instantaneous rate of change, which is the derivative.
• The limit process allows us to make calculations precise and accurate by avoiding direct division by zero. Instead, it captures what happens as we get infinitesimally close.

This concept is crucial for a deeper understanding of calculus.

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. In other words, they are fractions where both the numerator and the denominator are polynomials. The function from the exercise, \( r(s) = \frac{1}{s-2} \), is a simple rational function. Understanding rational functions is important when computing their derivatives, as they often involve additional algebraic manipulation.

• To find the derivative of a rational function using the definition, one must carefully manage algebraic expressions by finding common denominators and simplifying fractions.
• Once the algebraic simplifications are complete, the limit process can be applied to finalize the derivative calculation.
• Recognizing that rational functions are smooth except at points where the denominator is zero helps in analyzing their behavior and finding derivatives accurately.

Mastery of manipulating rational functions is key when you dive into calculus problems.

Calculus

Calculus is the branch of mathematics that studies how things change. It provides tools for modeling the physical world, predicting natural phenomena, and understanding the infinite intricacies of changing systems. The derivative is one of the two main concepts in calculus, the other being the integral.

• The derivative gives us a way to calculate rates of change and slopes of curves.
• Integral calculus, on the other hand, helps us find areas under curves and the accumulated quantities.
• In calculus, exploring the derivative involves a detailed understanding of the function and the limit process, while integration ties in cumulative effects.

Calculus opens doors to deeper insights in mathematics and various applications in real-life situations like physics, engineering, and economics.