Problem 11 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find \(D_{x} y\) using the rules of this section. $$ y=x^{2}+2 x $$

Step-by-Step Solution

Answer

The derivative is \( D_{x} y = 2x + 2 \).

Step 1: Identify the Function Components

First, identify the components of the function. The function given is \( y = x^2 + 2x \), which consists of two terms: \( x^2 \) and \( 2x \).

Step 2: Apply Power Rule to Each Term

The power rule states that if \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \). Using the power rule, the derivative of \( x^2 \) is \( 2x^{2-1} = 2x \). Similarly, for \( 2x \), the derivative is \( 2 \cdot x^{1-1} = 2 \).

Step 3: Combine the Derivatives

Add the derivatives of each term from Step 2 to find the derivative of the entire function. Therefore, the derivative \( D_{x} y = 2x + 2 \).

Key Concepts

Power Rule

The power rule is a cornerstone of calculus, especially useful when you're tackling polynomial functions. It’s a shortcut method that simplifies the process of finding derivatives, particularly for expressions involving powers of a variable. Here’s how it works:

• If you have a function of the form \(f(x) = x^n\), the derivative, according to the power rule, is \(f'(x) = nx^{n-1}\).
• This means you multiply the power of the variable by the coefficient (which is 1 if not shown), and then reduce the power by one.

Let's consider the term \(x^2\) from our example. Applying the power rule, we take the 2 in the exponent, bring it down as a coefficient, and subtract 1 from the original exponent, resulting in \(2x^{2-1} = 2x\).Similarly, a linear term like \(2x\) can be thought of as \(2x^1\). Using the rule here gives \(2 \cdot 1x^{1-1} = 2\), as any expression raised to the zero power simplifies to 1.The power rule quickly becomes intuitive with practice and is a vital tool for simplifying complex expressions in calculus.

Derivative Calculation

The process of derivative calculation is about finding the rate at which a function is changing at any given point. This involves differentiating, or finding the derivative of, a function. Here’s how you can think about derivative calculation:

• It is akin to measuring the slope of the tangent line to the function at a particular point, showing how the function's output changes with respect to changes in its input.
• Each term in a function is differentiated separately using the rules of differentiation, such as the power rule.

In our example function, \(y = x^2 + 2x\), we differentiate each term separately as follows:

• The derivative of \(x^2\) is \(2x\) as previously demonstrated using the power rule.
• The derivative of \(2x\) is simply 2, since the derivative of a constant multiplied by \(x\) results in that constant.

Lastly, to find the derivative of the whole function, you combine the results: \(D_x y = 2x + 2\).Understanding derivative calculation is essential in many fields such as engineering and physics, where it's used to model real-world phenomena.

Polynomial Functions

Polynomial functions are mathematical expressions consisting of variables raised to whole number powers and coefficients. A function of this type typically looks like \(f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where the coefficients \(a_n, a_{n-1}, \ldots, a_0\) are constants.

• These functions can include terms like \(x^2, 2x,\) or even higher powers like \(x^3\), all added or subtracted together.
• Crucially, polynomial functions are differentiable because their variables are raised to powers instead of being embedded in more complex expressions like fractions or roots.

In our example, \(y = x^2 + 2x\) is a simple polynomial with two terms:

• The \(x^2\) term represents a typical quadratic component.
• The \(2x\) term is linear, which makes the whole function easy and straightforward to differentiate.

Polynomial functions form the bedrock of calculus due to their simple differentiable nature, often serving as simplified models of more complex systems. Their straightforward rules of differentiation, like the power rule, make them a great starting point for learning to analyze more intricate relationships in math.