Problem 61 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Prove that the derivative of the function \(f(x)=|x|\) for \(x \neq 0\) is given by $$ f^{\prime}(x)=\left\\{\begin{array}{ll} 1 & \text { if } x>0 \\ -1 & \text { if } x<0 \end{array}\right. $$ Hint: Recall the definition of the absolute value of a number.

Step-by-Step Solution

Answer

To prove the derivative of the function \(f(x)=|x|\) for \(x \neq 0\),recall that the absolute value of a number can be described as a piecewise function: \[ f(x) =\left\\{\begin{array}{ll} x & \text { if } x>0 \\\ -x & \text { if } x<0 \end{array}\right. \] Find the derivative of each piece separately: - When \(x > 0\), the derivative is \(f'(x) = 1\). - When \(x < 0\), the derivative is \(f'(x) =-1\). Combining these, the derivative of the function \(f(x)=|x|\) for \(x \neq 0\) is: \[ f^{\prime}(x)=\left\\{\begin{array}{ll} 1 & \text { if } x>0 \\\ -1 & \text { if } x<0 \end{array}\right., \] which proves our result.

Step 1: Understand the absolute value function

The absolute value function is piecewise-defined as \(f(x) = x\) for \(x>0\) and \(f(x) = -x\) for \(x<0\). Remember that the derivative of a function gives us the slope at any point \(x\) on the function.

Step 2: Derive the left and right-sided equations separately

Since the function is defined differently for positive and negative \(x\), find the derivative of each separately. So, - When \(x > 0\), \(f(x) = x\), so \(f'(x) = 1\), as the derivative of \(x\) is equal to 1. - When \(x < 0\), \(f(x) = -x\), so \(f'(x) = -1\), as the derivative of \(-x\) is equal to -1.

Step 3: Combine Results

Now, we combine the results from step two to form a piecewise-defined function. The derivative of the function \(f^{\prime}(x)\) is given by: \[ f^{\prime}(x)=\left\\{\begin{array}{ll} 1 & \text { if } x>0 \\\ -1 & \text { if } x<0 \end{array}\right. \] Which is the expected result, hence, the derivative of the function \(f(x)=|x|\) for \(x \neq 0\) is proven.

Key Concepts

Absolute Value Function

When dealing with calculus, the absolute value function often presents itself as a challenge, primarily due to its unique shape and behavior. Defined simply, the absolute value of a real number is its distance from zero on the number line, without regard to direction. The function is expressed as \( f(x) = |x| \), and it is sharply pointed at the origin (\(x = 0\)), where it changes direction.

This sharp point is crucial because it indicates a change in the slope of the function. For values of \(x > 0\), the slope is continuously positive, and for \(x < 0\), it's continuously negative. To visualize this, imagine a V-shaped graph where the right arm is upward sloping, and the left arm is downward sloping. This peculiarity of the absolute value function will have significant implications when we try to calculate its derivative.

Piecewise-Defined Functions

The absolute value function is a classic example of a piecewise-defined function, which means it is defined by different expressions depending on the input value. In other words, the rules for computing the function's output change at certain points along the domain. For \( f(x) = |x| \), we have two rules:

• \( f(x) = x \) when \( x > 0 \)
• \( f(x) = -x \) when \( x < 0 \)

By defining the function in pieces, we can take into consideration the different behaviors for different portions of the domain. When we look at derivatives, piecewise definitions can help us find the slope of the curve in each section where the function’s formula is different.

Derivative Calculation

Calculating the derivative of a function means finding the rate at which the function's output changes with respect to its input. For smooth, continuous functions, this process is straightforward; however, for piecewise-defined functions like the absolute value function, we must be more careful.

The derivative tells us about the function's slope at every point. For the absolute value function, we already know the function behaves differently based on the domain. Thus, we calculate the derivative for each piece separately:

• When \(x > 0\), the derivative of \(f(x) = x\) is \(1\) because the graph is a straight line with a slope of \(1\).
• When \(x < 0\), the derivative of \(f(x) = -x\) is \(-1\) because here, the slope of the line is \(-1\).

So, the derivative of \(f(x)=|x|\) for \(x eq 0\) is a piecewise function as well, reflecting the two different slopes for positive and negative values of \(x\).

Calculus

Calculus is a branch of mathematics that focuses on change and motion. Through the operations of differentiation and integration, calculus provides powerful tools for analyzing and solving a myriad of problems in fields as diverse as physics, engineering, economics, and biology. The process of finding derivatives, a key part of calculus, helps us understand how a quantity changes in response to changes in another quantity.

The derivative is a fundamental concept in calculus because it gives us the instantaneous rate of change or the slope of the tangent line to the curve at any given point. With piecewise-defined functions like the absolute value, calculus allows us to dissect the function into understandable segments, so we can analyze and work with each piece individually. This method is particularly useful for understanding functions that have abrupt changes in behavior, as with the sharp point in the absolute value function.