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Chapter 2: Problem 1

find the second derivative of the function. $$ f(x)=9-2 x $$

Short Answer

Expert verified
The second derivative of the function \(f(x)=9-2x\) is \(0\).

Step by step solution

01

Find the first derivative

The first derivative of \(f(x)\), denoted as \(f'(x)\), can be found by applying the power rule of differentiation, which states that the derivative of \(x^n\) is \(nx^{n-1}\). Since \(f(x)=9-2x\) can be written as \(f(x)=9-2x^{1}\), the derivative, \(f'(x)\), is \(0 - 2*1*x^{1-1} = -2.\)
02

Find the second derivative

The second derivative of \(f(x)\), denoted as \(f''(x)\), is simply the derivative of the first derivative found in Step 1. Since the first derivative is a constant, \(-2\), the second derivative, \(f''(x)\), is \(0\), because the derivative of a constant is always \(0\).

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Key Concepts

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

Understanding the Power Rule
The power rule is a quick method to calculate derivatives of polynomial functions. Essentially, it tells us how to find the derivative when a function is in the form of \(x^n\). The rule states:
  • Differentiate \(x^n\) by bringing the power \(n\) in front and reducing the power by one, leading to \(nx^{n-1}\).
It's a simple yet powerful tool for any calculus student.
For our function, \(f(x) = 9 - 2x\), we apply the power rule to \(-2x^1\).
Calculating this, we get \(-2 \cdot 1 \cdot x^{1-1} = -2x^0 = -2\).
The constant 9 disappears because its derivative is zero. This technique will become second nature with practice.
Basics of Differentiation
Differentiation is the process of finding the derivative, which represents the instantaneous rate of change of a function.
This is highly important in mathematics, physics, economics, and beyond.
  • The derivative of a function gives us the slope of the tangent line at any point on the curve.
  • When dealing with polynomial functions like \(f(x) = 9 - 2x\), each term is treated individually.
In the first derivative, \(f'(x)\), we saw how \(-2x\) simplified to \(-2\).
This derivative was straightforward, but it illustrates how differentiation transforms functions into their rate of change.
Constant Function and Its Derivatives
A constant function is uniquely simple in differentiation. It doesn鈥檛 change, so its derivative is always zero.
  • When a function or term does not have a variable (like \(x\)), it remains constant.
  • For any constant \(c\), the derivative \(\frac{d}{dx}[c] = 0\).
In our problem, the first derivative \(f'(x) = -2\) is a constant function.
So, finding its derivative (the second derivative, \(f''(x)\)) gives us zero.
This tells us there's no curvature in the graph of \(f(x)\). It鈥檚 a straight line, confirming the lack of change in slope.

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