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Chapter 1: Problem 31

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

Short Answer

Expert verified
The derivative is .

Step by step solution

01

Identify the polynomial function

The given function is a polynomial: . This function has three terms: \(5x^{2}\), \(-7x\), and \(3\).
02

Differentiate each term separately

Utilize the power rule for differentiation, which states that : - For the first term, . - For the second term, . - For the third term, .
03

Combine the derivatives

Combine the derivatives found in Step 2: . Therefore, the derivative .

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Functions
In mathematics, a polynomial function is a type of function that is defined by a polynomial expression. This expression can have one or more terms, each consisting of a coefficient and a variable raised to a non-negative integer power.
For example, the polynomial function: \(5x^2 - 7x + 3\) has three terms:
  • \(5x^2\)
  • \(-7x\)
  • \(3\)
The coefficients are \(5\), \(-7\), and \(3\), and the powers of \(x\) are \(2\), \(1\), and \(0\), respectively.
This type of function is called a quadratic polynomial because the highest power of \(x\) in the expression is \(2\). Polynomial functions are important because they are simple to work with and can be used to approximate more complex functions.
Power Rule
The power rule is a basic but crucial concept in differentiation. It helps us find the derivative of a term where the variable \(x\) is raised to a power.
The power rule states: \(\frac{d}{dx}(x^n) = nx^{n-1}\) This means that you multiply the term by the power and then decrease the power by one.
Let's apply the power rule to each term in our polynomial function \(5x^2 - 7x + 3\):
  • For the term \(5x^2\), the derivative is \(5 \times 2x^{2-1} = 10x\)
  • For the term \(-7x\), the power of \(x\) is \(1\), so the derivative is \(-7 \times 1x^{1-1} = -7\)
  • For the term \(3\), which is a constant (can be written as \(3x^0\)), the derivative is \(0\)
The power rule is very helpful in differentiating polynomial functions because it allows us to differentiate each term separately and then combine the results.
Derivatives
Derivatives are fundamental in calculus as they represent the rate at which a function is changing at any given point. They are essentially the slope of the function at a particular point.
To find the derivative of a polynomial function like \(5x^2 - 7x + 3\), you need to differentiate each term separately using rules like the power rule.
From our previous steps, we know:
  • The derivative of \(5x^2\) is \(10x\)
  • The derivative of \(-7x\) is \(-7\)
  • The derivative of \(3\) is \(0\)
Combining these, we get the derivative of the entire function:
\(\frac{d}{dx}(5x^2 - 7x + 3) = 10x - 7\)
This tells us how the function \(5x^2 - 7x + 3\) changes as \(x\) changes. Derivatives are used in various fields like physics, engineering, and economics to model change and predict future behavior.

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Most popular questions from this chapter

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