Problem 30 - Calculus and its applications

Question

Find each derivative. $$ \frac{d}{d x}\left(5 x^{2}-7 x+3\right) $$

Step-by-Step Solution

Answer

The derivative is \( 10x - 7 \).

Step 1: Identify the function and the rule to apply

The function given is a polynomial: \( 5x^2 - 7x + 3 \). We will use the power rule for differentiation, which states that the derivative of \( ax^n \) is \( n \cdot ax^{n-1} \).

Step 2: Differentiate the first term

Apply the power rule to the first term \( 5x^2 \). The exponent is 2, so the derivative is \( 2 \cdot 5x^{2-1} = 10x \).

Step 3: Differentiate the second term

Apply the power rule to the second term \( -7x \). The exponent is 1, so the derivative is \( 1 \cdot -7x^{1-1} = -7 \).

Step 4: Differentiate the third term

The third term is a constant \( 3 \). The derivative of a constant is 0, so the third term's derivative is 0.

Step 5: Combine all derivatives

Combine the derivatives of each term: \( 10x - 7 + 0 \). This simplifies to \( 10x - 7 \).

Key Concepts

Understanding the Power Rule

The power rule is a fundamental tool in differential calculus used to find the derivative of polynomial expressions. It allows us to easily differentiate terms where variables are raised to a power.
For instance, the power rule states that if we have a term of the form \( ax^n \), its derivative is calculated as \( n \cdot ax^{n-1} \). Here is how it works:

• Identify the Exponent: The exponent \( n \) is key. It shows how many times the base \( x \) is used as a factor.
• Apply the Rule: Multiply the entire term by \( n \) and reduce the exponent by 1. This gives the new power of \( x \).

This straightforward rule helps in reducing complexities when dealing with longer equations. It is a quick way to find how functions change at any point.

What is a Polynomial Function?

A polynomial function is a mathematical expression involving a sum of powers of a variable, each multiplied by coefficients.
Poly means 'many', and nomial refers to 'terms'. Hence, a polynomial is a combination of multiple terms. Each of these powers is a whole number.

• Form: A polynomial in \( x \) can be expressed as \( a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \), where each \( a_i \) is a coefficient and \( n \) is a nonnegative integer.
• Degree: The degree of a polynomial is the highest power of the variable. For example, in \( 5x^2 - 7x + 3 \), the degree is 2.

Polynomials are extensively used due to their simple structure and easy manipulation through arithmetic operations and calculus techniques.

Differentiation Steps for Polynomials

To differentiate polynomials, investigate each term individually and apply the power rule. Follow a systematic approach:

• Identify Each Term: Break down the polynomial into its individual terms. For instance, separating the expression \( 5x^2 - 7x + 3 \) into terms: \( 5x^2 \), \( -7x \), and \( 3 \).
• Apply the Power Rule: Use the power rule on each term. For \( 5x^2 \), differentiate to get \( 10x \). For \( -7x \), the derivative is \( -7 \). For the constant term \( 3 \), its derivative is 0 because constants disappear on differentiation.
• Combine Derivatives: After differentiating each term, combine them. The solution for our example simplifies to \( 10x - 7 \) after summing up individual derivatives.

By taking these steps, you ensure that each aspect of the polynomial is addressed correctly. This methodical process builds a foundational understanding of calculus and ensures functional accuracy.