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Chapter 3: Problem 16

Find the derivative. Assume \(a, b, c, k\) are constants. $$y=4.2 q^{2}-0.5 q+11.27$$

Short Answer

Expert verified
The derivative is \(8.4q - 0.5\).

Step by step solution

01

Apply the Power Rule

The power rule for differentiation states that the derivative of a term of the form \(ax^n\) is \(n \cdot ax^{n-1}\). Apply this rule to the first term \(4.2q^2\). The derivative is \(2 \cdot 4.2q^{2-1} = 8.4q\).
02

Differentiate the Linear Term

Differentiate the second term \(-0.5q\) according to the power rule where the derivative of \(bx\) is \(b\). Here, the derivative is \(-0.5\).
03

Differentiate the Constant

The derivative of a constant is always 0. Therefore, the derivative of \(11.27\) is 0.
04

Combine the Derivatives

Combine the derivatives from Steps 1, 2, and 3. The overall derivative of the function is \(8.4q - 0.5\).

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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 for finding the derivative of polynomial terms. It simplifies the process by allowing you to directly differentiate terms of the form \( ax^n \). The rule states that the derivative of \( ax^n \) is \( n \cdot a x^{n-1} \). This means:
  • You multiply the exponent \( n \) by the coefficient \( a \).
  • You then reduce the exponent by one.
Let's see how this applies to our given exercise:For the term \( 4.2q^2 \), the exponent \( n \) is 2. So, multiply 2 by the coefficient 4.2, resulting in \( 8.4 \). Then, lower the power of \( q \) by one to obtain \( q^{1} \).
Thus, the derivative of \( 4.2q^2 \) is \( 8.4q \).
Power Rule makes handling higher power terms easier, especially in polynomial functions.
Linear Term Differentiation
Differentiating linear terms is straightforward. A linear term is a first-degree polynomial, usually written as \( bx \), where \( b \) is a constant. The derivative of such a term is simply the constant \( b \), because:
  • The power of \( x \) in a linear term is 1.
  • Using the power rule, the derivative is \( 1 \cdot b x^{1-1} = b \).
In our example, the linear term is \( -0.5q \). By applying linear term differentiation, the derivative becomes \( -0.5 \), since \( q \) has no power attached (or it could be considered as \( q^1 \)). Linear terms, thanks to this simplification, are the easiest to differentiate.
Constant Term Differentiation
Constant terms do not involve a variable; they are just numbers. When differentiating a constant, you apply the rule that the derivative of any constant is zero. Here's why:
  • A constant can be viewed as a term with a variable raised to the power of 0.
  • Using the formula \( ax^0 \), differentiation results in \( 0 \cdot ax^{-1} = 0 \).
In our equation, \( 11.27 \) is the constant term. Differentiating this yields 0, indicating that constants don't change with respect to the variable. Differentiating constants underscores how calculus focuses largely on variable changes. By recognizing this, you can quickly identify and differentiate constant terms in any function.

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