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Chapter 2: Problem 16

Find the second derivative. $$ g(x)=m x+b ; m, b \text { are constants } $$

Short Answer

Expert verified
The second derivative \(g''(x)\) is 0.

Step by step solution

01

Identify the Function

Given the function is \(g(x) = mx + b\), where \(m\) and \(b\) are constants.
02

Calculate the First Derivative

To find the first derivative of \(g(x) = mx + b\), differentiate the function with respect to \(x\). Since \(m\) and \(b\) are constants, the derivative of \(mx\) is \(m\), and the derivative of a constant \(b\) is 0. Thus, the first derivative is: \[ g'(x) = m \]
03

Calculate the Second Derivative

To find the second derivative, differentiate the first derivative \(g'(x) = m\) with respect to \(x\). Since \(m\) is a constant, the derivative of a constant is 0. Thus, the second derivative is: \[ g''(x) = 0 \]

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Key Concepts

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

First Derivative
To understand the first derivative, let's start with the basics. The first derivative of a function tells us how the function's output changes as the input changes. Essentially, it's the rate of change or the slope of the function at any given point. When we differentiate a function, we are finding its first derivative.

In the given exercise, the function is: \[ g(x) = mx + b \]
Here, 'm' and 'b' are constants. To differentiate this function, we apply the basic rules of differentiation:
  • The derivative of a constant is 0.
  • The derivative of mx is just m, considering 'm' as a constant multiplier.

Therefore, the first derivative of our function is: \[ g'(x) = m \] This means that the slope of the function \(g(x)\) is constant and equal to 'm'.
Constants in Calculus
Constants play a crucial role in calculus. They are numbers that do not change; they remain fixed. When differentiating functions, it's important to know how to handle constants.

For any constant 'k': \[ \frac{d}{dx}(k) = 0 \]
In our exercise, 'm' and 'b' are constants. When we differentiated \(g(x) = mx + b\), the constants affected the outcome of the derivatives:
  • 'm' multiplied by 'x' resulted in 'm' when differentiated.
  • 'b' remained unaffected and became 0 when differentiated.

Understanding how to deal with constants is key in making the differentiation process clearer and more efficient.
Differentiation
Differentiation is a fundamental concept in calculus. It involves finding the derivative of a function, which represents the rate of change of the function with respect to one of its variables.

To differentiate a function means to apply specific rules to find its derivative. Below are some key rules for basic differentiation:
  • The power rule: \( \frac{d}{dx}(x^n) = nx^{n-1} \)
  • The constant rule: \( \frac{d}{dx}(c) = 0 \) for any constant 'c'.
  • The sum rule: \( \frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x) \)
  • The product rule: \( \frac{d}{dx} [f(x)g(x)] = f'(x)g(x) + f(x)g'(x) \)
  • The quotient rule: \( \frac{d}{dx} \frac{f(x)}{g(x)} = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2} \)

In our specific exercise, understanding these rules allowed us to correctly differentiate \(g(x) = mx + b\). From the first derivative to the second derivative, the principles of differentiation guided us to the solution.

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