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Chapter 2: Problem 13

Find the derivative of each function. $$ f(x)=4 x^{2}-3 x+2 $$

Short Answer

Expert verified
The derivative is \( f'(x) = 8x - 3 \).

Step by step solution

01

Identify the Function

The function given is \( f(x) = 4x^2 - 3x + 2 \), which is a polynomial in the standard form.
02

Apply the Power Rule

The power rule states that the derivative of \( ax^n \) is \( n \cdot ax^{n-1} \). Begin applying this rule to each term of the polynomial.
03

Differentiate the First Term

The first term is \( 4x^2 \). Applying the power rule: the derivative is \( 2 \cdot 4x^{2-1} = 8x \).
04

Differentiate the Second Term

The second term is \( -3x \). According to the power rule, the derivative is \( 1 \cdot (-3)x^{1-1} = -3 \).
05

Differentiate the Constant Term

The third term is a constant \( 2 \). The derivative of any constant is \( 0 \).
06

Combine the Derivatives

Combine the derivatives of each term: \( 8x \) from the first term, \( -3 \) from the second term, and \( 0 \) from the third term. The derivative of the function is \( f'(x) = 8x - 3 \).

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Key Concepts

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

Power Rule
The power rule is a fundamental tool in calculus used to quickly find the derivatives of polynomial terms. It states that if you have a term in the form of \( ax^n \), then the derivative is calculated as \( n \cdot ax^{n-1} \). This rule simplifies the process of differentiation, making it straightforward to handle expressions involving powers of \( x \).
  • The exponent \( n \) becomes a coefficient, multiplying the existing coefficient \( a \).
  • Subtract 1 from the exponent \( n \) to find the new power of \( x \).

Let's see this in action: If you have \( 5x^3 \), using the power rule, the derivative becomes \( 3 \cdot 5x^{3-1} = 15x^2 \). This makes finding derivatives of polynomial functions easy and quick!
Polynomial Function
A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In the given exercise, the function \( f(x) = 4x^2 - 3x + 2 \) is a classic example of a polynomial.
  • Each term of the polynomial has a coefficient and a variable raised to an exponent.
  • In this case, the degrees of the terms range from 0 to 2.

Polynomials are major players in various fields such as algebra, calculus, and engineering because they can model a wide range of real-world situations. Their properties, such as continuity and smoothness, make them ideal for analysis using calculus tools like derivatives and integrals.
Differentiation Steps
Differentiating a polynomial involves a series of straightforward steps, especially when utilizing the power rule. For the function \( f(x) = 4x^2 - 3x + 2 \), here's how it breaks down:
  • Identify the Function: This step ensures you understand what you are working with and guides your application of calculus rules.
  • Apply the Power Rule: For each term \( ax^n \), use \( n \cdot ax^{n-1} \) to find the derivative.
  • Handle Each Term: - First, differentiate \( 4x^2 \) to get \( 8x \). - Then, differentiate \( -3x \) to get \( -3 \) (remember, \( x^1 \) reduces to \( x^0 \) which is 1). - Constants like \( +2 \) turn into 0 because their rate of change is zero.
  • Combine the Results: Sum the derivatives of each term to find the overall derivative of the polynomial: \( f'(x) = 8x - 3 \).

These steps illustrate how differentiation transforms a polynomial function, revealing how each part contributes to its changing slope. Practice with different functions hones your calculus skills and makes tackling more complex equations a manageable task.

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Most popular questions from this chapter

Determine whether each function is continuous or discontinuous. If discontinuous, state where it is discontinuous. $$ \begin{array}{l} f(x)=\left\\{\begin{array}{ll} 3-x & \text { if } x \leq 4 \\ 10-2 x & \text { if } x>4 \end{array}\right.\\\ \text { [Hint: See Exercise 37.] } \end{array} $$

For each function: a. Find \(f^{\prime}(x)\) using the definition of the derivative. b. Explain, by considering the original function, why the derivative is a constant. $$ f(x)=3 x-4 $$

Find the derivative of \(\left(x^{2}+1\right)^{2}\) in two ways: a. By the Generalized Power Rule. b. By "squaring out" the original expression and then differentiating. Your answers should agree.

For each function: a. Find \(f^{\prime}(x)\) using the definition of the derivative. b. Explain, by considering the original function, why the derivative is a constant. $$ f(x)=2 x-9 $$

Suppose that \(L(x)\) is a function such that \(L^{\prime}(x)=\frac{1}{x} .\) Use the Chain Rule to show that the derivative of the composite function \(L(g(x))\) is \(\frac{d}{d x} L(g(x))=\frac{g^{\prime}(x)}{g(x)} .\)
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