Problem 49 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In Exercises 49-52, find the third derivative of the given function. \(f(x)=3 x^{4}-4 x^{3}\)

Step-by-Step Solution

Answer

The third derivative of the given function is \(f'''(x) = 72x - 24\).

Step 1: Find the first derivative

Apply the power rule for differentiation, which is: \(\frac{d}{dx} (x^n) = nx^{(n-1)}\) $$ \frac{d}{dx}(3x^4 - 4x^3) = \frac{d}{dx}(3x^4) - \frac{d}{dx}(4x^3) = 3\cdot4x^{(4-1)} - 4\cdot3x^{(3-1)} $$ So, \(f'(x) = 12x^3 - 12x^2\).

Step 2: Find the second derivative

Now, find the second derivative by applying the power rule again on the first derivative. $$ \frac{d^2}{dx^2}(12x^3 - 12x^2) = \frac{d}{dx}(12x^3) - \frac{d}{dx}(12x^2) = 12\cdot3x^{(3-1)} - 12\cdot2x^{(2-1)} $$ So, \(f''(x) = 36x^2 - 24x\).

Step 3: Find the third derivative

Finally, find the third derivative by applying the power rule once more on the second derivative. $$ \frac{d^3}{dx^3}(36x^2 - 24x) = \frac{d}{dx}(36x^2) - \frac{d}{dx}(24x) = 36\cdot2x^{(2-1)} - 24\cdot1x^{(1-1)} $$ So, \(f'''(x) = 72x - 24\). The third derivative of the given function is \(f'''(x) = 72x - 24\).

Key Concepts

Power Rule for Differentiation

Mastering the power rule for differentiation is essential when delving into calculus and tackling problems involving rates of change. Imagine the power rule as a handy shortcut that bypasses longer, more tedious methods of finding derivatives; it streamlines calculations significantly.

So, how does it work? The power rule is applied to functions where the variable, typically denoted as x, is raised to a power, symbolized as n. The rule states that to find the derivative of such a function, you multiply the exponent n by the coefficient of the term and reduce the exponent by one. Mathematically, it is expressed as follows:
\[\begin{equation}\frac{d}{dx}(ax^n) = n \times ax^{(n-1)}\end{equation}\]
For instance, if you have a term like \(5x^3\), applying the power rule would give you \(15x^2\), since you multiply the exponent, 3, by the coefficient, 5, and then subtract one from the exponent. This rule provides a quick and reliable way to move forward in finding the first, second, and even higher order derivatives of polynomials.

First Derivative

In calculus, the first derivative is much like the opening act of an analysis performance—setting the stage for understanding the behavior of functions. Broadly speaking, the first derivative of a function represents the rate at which the function's value changes with respect to a change in its input value.

This mathematical tool allows us to determine the slope of the tangent line to the function's graph at any given point. In simpler terms, it answers the question: 'At a specific instant, how fast is the function's value changing?' When the first derivative of a function is positive, the function is increasing, and when it's negative, the function is decreasing. A zero value indicates a potential maximum or minimum point, or a point of inflection.

Applying the First Derivative

In the exercise provided, we used the power rule to find the first derivative of the polynomial \(3x^4 - 4x^3\). Through each term's differentiation, we got \(f'(x) = 12x^3 - 12x^2\). This result reveals the instantaneous rate of change of the function and prepares us for further analysis like finding the second derivative or determining the function’s critical points.

Second Derivative

The second derivative isn’t merely a follow-up to the first; it's a profound expansion of the analysis, revealing the curvature and concavity of the original function. Technically, it's the derivative of the derivative, so you're taking the rate of change of the rate of change.

One can interpret the second derivative, \(f''(x)\), as describing the acceleration of the function's value—how the slope itself is changing. A positive second derivative indicates the function is concave up (like a cup), meaning it's accelerating upwards, while a negative second derivative shows that the function is concave down, accelerating downwards. When the second derivative is zero, it may indicate a point of inflection, where the function changes concavity.

Calculating the Second Derivative

Using the power rule again on the first derivative \(f'(x) = 12x^3 - 12x^2\), yielded the second derivative \(f''(x) = 36x^2 - 24x\). This step provided us with an understanding of how the slope of the tangent line is changing as x varies, crucial for determining the nature of the graph of the original function.