Problem 22 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=\frac{2}{3 x}-\frac{2}{3} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the terms to differentiate

We need to find \( D_x y \) for the function \( y = \frac{2}{3x} - \frac{2}{3} \). This consists of two terms: \( \frac{2}{3x} \) and \( -\frac{2}{3} \). Each term will be differentiated separately.

Step 2: Differentiate the first term \( \frac{2}{3x} \)

Rewrite \( \frac{2}{3x} \) as \( \frac{2}{3} \cdot \frac{1}{x} = \frac{2}{3} x^{-1} \). Using the power rule, \( \frac{d}{dx}(x^n) = nx^{n-1} \), we get \( \frac{d}{dx}(x^{-1}) = -x^{-2} \). Therefore, \( \frac{d}{dx} \left( \frac{2}{3}x^{-1} \right) = \frac{2}{3}(-x^{-2}) = -\frac{2}{3x^2} \).

Step 3: Differentiate the second term \(-\frac{2}{3}\)

The derivative of a constant is zero. Therefore, \( \frac{d}{dx}(-\frac{2}{3}) = 0 \).

Step 4: Combine the derivatives

Add the derivatives of the individual terms to get the derivative of the whole expression: \( D_x y = -\frac{2}{3x^2} + 0 = -\frac{2}{3x^2} \).

Key Concepts

Power Rule

The power rule is a basic principle in the differentiation of functions. It's a formula to find the derivative of a term when it has a power. Think of a function like this: - If you have a function represented as \( x^n \), you differentiate it with respect to \( x \) using the power rule: \( \frac{d}{dx}(x^n) = nx^{n-1} \). This means that you will multiply the function by its exponent and subtract one from the exponent.
- Consider the example \( x^{3} \). Its derivative is \( \frac{d}{dx}(x^3) = 3x^{2} \). This simple formula is incredibly useful for finding derivatives of polynomials and similar expressions, helping you find how the function changes or its rate of change.
- In our exercise, we turned \( \frac{2}{3x} \) into \( \frac{2}{3} x^{-1} \). Using the power rule, it becomes \(-\frac{2}{3}x^{-2} \) after differentiation.

Constant Functions

The simplest functions to differentiate are constant functions. A constant function is one where the output value is the same no matter the input. An example is \( y = c \), where \( c \) is a constant.
- The derivative of a constant is always zero. This is because a constant function doesn't change, so its rate of change is zero. Mathematically, \( \frac{d}{dx}(c) = 0 \).
- In the exercise, the term \(-\frac{2}{3}\) is a constant, so its derivative is zero. Constant functions often appear in various expressions, and knowing their derivative is always zero can simplify the differentiation process.

Finding Derivatives

Finding derivatives is a critical process in calculus that measures how a function changes as its input changes. It involves applying rules to derive the slope or rate of change of a function.
- Derivatives can help us understand the behavior of functions, such as finding where a function increases or decreases, or identifying maximum and minimum points.
- To find the derivative, identify each term in a function and apply the appropriate differentiation rules, like the power rule for polynomials or acknowledging constants when they appear. - Once you've differentiated each part separately, combine them to find the derivative of the entire function. In the exercise presented, this Series of steps led us to find that the derivative \( D_x y = -\frac{2}{3x^2} \), thus obtaining crucial information about the function's behavior and variation.