Problem 22 - Calculus an Applied Approach

Question

find the third derivative of the function. $$ f(x)=\frac{1}{x} $$

Step-by-Step Solution

Answer

The third derivative of the function \(f(x)=\frac{1}{x}\) is \(f'''(x)=-6*\frac{1}{x^4}\).

Step 1: Identifying the function

The given function is \(f(x)=\frac{1}{x}\), which can be rewritten as \(f(x)=x^{-1}\) for simplicity.

Step 2: Differentiation of the function

Apply the power rule of differentiation, which says that the derivative of \(x^n\) is \(n*x^{n-1}\). This rule is applied here, where \(n=-1\). The first derivative of \(f(x)\) is obtained as \(f'(x)=-1*x^{-2}=-\frac{1}{x^2}\).

Step 3: Differentiation of the first derivative

Again applying the power rule of differentiation to \(f'(x)=-\frac{1}{x^2}\), or \(f'(x)=-1*x^{-2}\), where \(n=-2\). The second derivative of \(f(x)\) can be obtained as \(f''(x)=2*x^{-3}=2*\frac{1}{x^3}\).

Step 4: Differentiation of the second derivative

Again applying the power rule of differentiation to \(f''(x)=2*\frac{1}{x^3}\), or \(f''(x)=2*x^{-3}\), where \(n=-3\). The third derivative of \(f(x)\) is obtained as \(f'''(x)=-6*x^{-4}=-6*\frac{1}{x^4}\).

Key Concepts

Power Rule of Differentiation

One of the most fundamental tools in calculus is the power rule of differentiation. This rule makes it easier to find the derivative of a function where the variable, usually denoted as 'x', is raised to a power. The basic formula for the power rule is as follows: if you have a function in the form of \(f(x) = x^n\), the derivative, denoted as \(f'(x)\), is \(n \cdot x^{n-1}\).

Let's take a closer look at how it applies to the function \(f(x)=x^{-1}\), which is the same as \(1/x\). Following the power rule, the first derivative is \(-1 \cdot x^{-1-1}\), which simplifies to \(-x^{-2}\) or \(-1/x^2\). This process can be repeated for higher derivatives, as each subsequent derivative will involve reducing the exponent by one and multiplying by the new exponent.

First Derivative

The first derivative of a function represents the rate at which the function's value changes with respect to the variable. In more practical terms, it's often described as the slope of the tangent line to the function's graph at any given point. In the context of our example, the first derivative of the function \(f(x) = x^{-1}\) is found using the power rule and is given by \(f'(x) = -x^{-2}\) or \(-1/x^2\).

This derivative shows how rapidly the function \(f(x) = 1/x\) is changing at any point 'x'. If you graphed the function \(f(x)\) and its derivative \(f'(x)\), you would see that the slope of the tangent line becomes steeper as 'x' approaches zero from either direction, indicating an increase in the rate of change.

Second Derivative

Continuing with successive differentiations, we reach the second derivative, which is essentially the derivative of the first derivative. It offers insights into the concavity and inflection points of the original function. If we apply the power rule to our first derivative \(-x^{-2}\), we obtain the second derivative \(f''(x) = 2x^{-3}\) or \(2/x^3\).

The second derivative tells us about the curvature of the function's graph. In our example \(f''(x)\), a positive value suggests the graph of the function is concave up, shaped like a cup, while a negative value indicates it is concave down, like an upside-down cup. This helps in understanding the nature of the function's graph and predicting the behavior of the function.

Function Differentiation

Function differentiation is the process of finding the derivative, or the rate of change, of a function. This process often begins with the power rule (as discussed earlier) but encompasses other techniques for more complex functions. Differentiation enables the analysis of functions and is crucial in many applications in physics, engineering, economics, and beyond.

In differentiating the function \(f(x) = 1/x\), we applied the power rule three times successively to calculate the first, second, and eventually, the third derivative. The third derivative, or the rate at which the second derivative changes, is \(-6x^{-4}\) or \(-6/x^4\). While higher derivatives become less common in basic applications, they can still provide deeper insights into the function's behavior and are used in more advanced studies like series expansions and differential equations.