Question

In Exercises, find the second derivative of the function. $$ f(x)=4 x+15 $$

Short answer

The second derivative of the function \(f(x)=4x+15\) is \(0\).

Step-by-step solution

Find the First Derivative

The derivative of a constant times a variable is the constant, and the derivative of a constant is zero. So, the first derivative of \(f(x)\) is The derivative of \(4x\) is \(4\) and the derivative of \(15\) is \(0\). So, \(f'(x)=4\).

Find the Second Derivative

The second derivative is the derivative of the first derivative. Since the first derivative \(f'(x)\) is a constant (\(4\)), its derivative is \(0\). So, the second derivative \(f''(x)\) is \(0\).

Key concepts

First Derivative

Understanding the first derivative is crucial in calculus. It represents the rate at which a function is changing at any given point on its graph. For the function f(x) = 4x + 15, the first derivative is found by applying the basic rules of differentiation. Any constant multiplied by a variable, like 4x, is differentiated to the constant itself. This is because the rate at which 4x increases is uniformly 4. Hence, in our example, the rate of change, or the first derivative, is simply 4.

This derivative signifies that for any unit increase in x, the function f(x) increases by 4 units. Understanding this concept allows students to analyze and predict the behavior of linear functions effectively.

Derivative of a Constant

When you encounter a constant in a function, like the 15 in f(x) = 4x + 15, its derivative is always 0. This is because a constant doesn't change, it doesn't grow or shrink as x changes; its rate of change is zero. Mathematically speaking, this is because a constant has no variable component that could respond to changes in x. In calculus education, emphasizing this fundamental rule helps demystify why constants 'disappear' when differentiating. Though it may seem minor, the derivative of a constant is a foundational concept that solidifies a student's understanding of differentiation.

Calculus Education

Calculus can be daunting for many students, but breaking it down into comprehensible pieces can ease the learning curve. It starts with the fundamental concepts like derivatives and understanding their geometrical and physical interpretations. Exercises that involve finding derivatives step by step help students internalize these concepts. For instance, working through the function f(x) = 4x + 15 is a good practice for beginners as it involves both a variable component and a constant. By scaffolding learning—starting with simple functions and moving to more complex examples—educators can help students build the confidence and skills needed for calculus success.

Derivative of a Linear Function

Linear functions are the simplest form of polynomials, and they have the general form f(x) = mx + b, where m is the slope and b is the y-intercept. The derivative of a linear function is always the slope, m. In the example f(x) = 4x + 15, 4 is the slope, so the first derivative f'(x) is 4. This consistency is a powerful tool in calculus, allowing students to quickly identify the rate at which the function changes without complex calculations. Recognizing that the derivative of a linear function provides direct insight into the function's steepness and direction is key to mastering the basics of differential calculus.