Question

Find \(D_{x} y\) using the rules of this section. $$ y=\frac{2}{3 x}-\frac{2}{3} $$

Short answer

The derivative is \( D_{x} y = -\frac{2}{3x^2} \).

Step-by-step solution

Understand the Expression

The given function is \[ y = \frac{2}{3x} - \frac{2}{3}. \] This function includes terms that are both a rational function and a constant.

Differentiate using the Quotient Rule

For the term \( \frac{2}{3x} \), consider it as \( \frac{2}{3} \times \frac{1}{x} \). The derivative of \( \frac{1}{x} \) with respect to \( x \) is \( -\frac{1}{x^2} \). Thus, \[ D_{x}\left( \frac{2}{3} \cdot \frac{1}{x} \right) = \frac{2}{3} \times \left( -\frac{1}{x^2} \right) = -\frac{2}{3x^2}. \]

Differentiate the Constant Term

For the constant term \( \frac{2}{3} \), its derivative is zero since the derivative of a constant is always zero. Thus,\[ D_{x} \left( -\frac{2}{3} \right) = 0. \]

Combine the Derivatives

Combine the derivatives from step 2 and step 3: \[ D_{x}y = -\frac{2}{3x^2} + 0. \] This simplifies to \[ D_{x} y = -\frac{2}{3x^2}. \]

Key concepts

Quotient Rule

When finding the derivative of a rational function, the Quotient Rule is one of the essential tools in calculus. A rational function is a ratio of two polynomials. Sometimes, dividing these polynomials is not straightforward, so the Quotient Rule comes in handy. The rule states that if you have a function \( f(x) = \frac{u(x)}{v(x)} \), the derivative, \( f'(x) \), is given by:\[ f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{v(x)^2}\]This rule effectively tells us how to handle division in derivatives. Instead of finding the derivative of the entire quotient at once, it breaks it down into parts involving the derivatives of the numerator \( u(x) \) and the denominator \( v(x) \). This technique is particularly useful since dealing with each component separately often makes the process simpler. In our example, although we don't explicitly apply the full Quotient Rule, understanding this rule is essential when transforming terms like \( \frac{1}{x} \) or in more complex rational expressions.

Derivative of Constants

In calculus, understanding the derivative of constants is fundamental. A constant is a term that does not change or depend on any variables. For any constant \( c \), the derivative with respect to any variable \( x \) is always zero. The reasoning for this is quite logical:- Imagine a flat line (e.g., \( y = c \)). The slope of this line is zero because it doesn't rise or fall—it's constant.- In mathematical terms, the derivative represents the rate of change, and a constant doesn't change, so that rate is zero.This concept simplifies calculations a lot. In the context of our exercise, for instance, the constant term \( -\frac{2}{3} \) becomes irrelevant when we take the derivative. We immediately know it contributes nothing and can be set aside from the computations.

Rational Functions

Rational functions are the division of one polynomial by another, for example, \( \frac{2}{3x} \). These types of functions are fundamental in calculus due to their prevalence in various applications and problems.Key characteristics include:

• They can have undefined points where the denominator equals zero, leading to vertical asymptotes.
• They can be expressed as simpler forms if the numerator or denominator can be factored and reduced.
• This form often requires specific rules (like the Quotient Rule) or simplifications for finding derivatives.

For the example \( y = \frac{2}{3x} \), we treated it as \( \frac{2}{3} \times \frac{1}{x} \) and found its derivative separately. Understanding these concepts is crucial, as the manipulation and differentiation of rational functions appear frequently in mathematical problems and real-world applications.