Problem 71 - University Calculus: Early Transcendentals

Question

The general polynomial of degree \(n\) has the form $$ P(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{2} x^{2}+a_{1} x+a_{0}$$ where \(a_{n} \neq 0 .\) Find \(P^{\prime}(x)\).

Step-by-Step Solution

Answer

The derivative is \(P'(x) = n \cdot a_{n} x^{n-1} + (n-1) \cdot a_{n-1} x^{n-2} + \cdots + 2 \cdot a_{2} x + a_{1}.\)

Step 1: Understanding the Derivative

The derivative of a function gives us the rate at which the function values are changing. For a polynomial, this involves applying the power rule to each term separately.

Step 2: Apply the Power Rule

The power rule states that the derivative of \(a_{n}x^n\) is \(n \cdot a_{n} \cdot x^{n-1}\). We'll apply this rule to each term of the polynomial \(P(x)\).

Step 3: Differentiate Each Term of the Polynomial

Differentiate each term: - The derivative of \(a_{n} x^{n}\) is \(n \cdot a_{n} x^{n-1}\).- The derivative of \(a_{n-1} x^{n-1}\) is \((n-1) \cdot a_{n-1} x^{n-2}\).- Continue this pattern down to the constant term \(a_0\), which becomes 0 because the derivative of a constant is zero.

Step 4: Combine the Differentiated Terms

Combine all the differentiated terms to form the derivative of the polynomial: \[ P'(x) = n \cdot a_{n} x^{n-1} + (n-1) \cdot a_{n-1} x^{n-2} + \cdots + 2 \cdot a_{2} x + a_{1}. \]

Key Concepts

Power Rule

In the world of calculus, the power rule is a handy tool for finding the derivative of a polynomial term. This rule applies to terms with the form \(a \cdot x^n\), where \(a\) is a constant and \(n\) is a positive integer. The power rule dictates that the derivative of such a term is \(n \cdot a \cdot x^{n-1}\).

• The exponent \(n\) becomes a coefficient in front of the term after differentiation.
• The new exponent of \(x\) becomes \(n-1\) as you decrease it by one.

Let's say you have \(5x^4\). By applying the power rule:

• Bring down the exponent 4 to multiply it by 5.
• Reduce the exponent of \(x\) by 1, resulting in \(20x^3\).

Degree of Polynomial

The degree of a polynomial is the largest exponent in the polynomial expression. It tells us the highest power to which the variable is raised. For example, in the polynomial \(3x^5 + 4x^3 + x + 7\), the degree is 5 because the term \(3x^5\) has the largest exponent.
A polynomial of degree \(n\) will have terms ranging from \(x^n\) down to possibly a constant term. Identifying the degree is crucial because:

• It determines the behavior of the function, especially as \(x\) becomes very large or very small.
• The degree also guides us on how many times we might need to apply the power rule during differentiation.

Differentiation

Differentiation is the process used to find the derivative of a function. This process gives the rate at which a function is changing at any point. For polynomials, differentiation involves using the power rule to obtain a new expression called the derivative. This technique is straightforward due to its repetitive nature:

• Each term of the polynomial is considered independently.
• Apply the power rule to each term by moving the exponent forward and reducing it by one.
• Combine all the differentiated terms to obtain the result.

This process transforms the original polynomial, giving you a new polynomial which tells you the slope or rate of change at any point on the graph of the function.

Constant Term Derivative

In the context of polynomial differentiation, encountering a constant term is common. A constant term is a number with no variable attached. When differentiating, the derivative of a constant is zero. This is because constants do not change, having no slope.

• A constant like \(7\) differentiates to \(0\).
• This zero is why constant terms disappear in the differentiation process.

Understanding why this happens is simple:

• Since a derivative represents a rate of change, and constants do not change, their rate is zero.
• Think about the graph of a constant function: it's a flat horizontal line.
• The slope of this line is zero, reinforcing that the derivative of a constant is zero.