Finding The Second Derivative Of F(x) = 12x² - 4x + 7 + Ln(x)

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In the realm of calculus, derivatives play a pivotal role in understanding the behavior of functions. They provide insights into the rate of change of a function, its increasing and decreasing intervals, concavity, and points of inflection. Among these, the second derivative, denoted as f''(x), holds particular significance as it reveals the concavity of the function's graph. A positive second derivative indicates that the function is concave up, resembling a smile, while a negative second derivative signifies that the function is concave down, resembling a frown. In this comprehensive exploration, we embark on a journey to unravel the second derivative of the function f(x) = 12x² - 4x + 7 + ln(x), delving into the step-by-step process and highlighting the underlying principles.

Step 1: Finding the First Derivative, f'(x)

To embark on our quest for the second derivative, the first step involves determining the first derivative, f'(x). The first derivative, as mentioned earlier, provides valuable information about the function's rate of change. To find f'(x), we apply the fundamental rules of differentiation to each term of the function f(x) = 12x² - 4x + 7 + ln(x). Let's dissect the process:

  1. Differentiating 12x²: The power rule of differentiation dictates that the derivative of x^n is nx^(n-1). Applying this to 12x², we get 12 * 2x^(2-1) = 24x.
  2. Differentiating -4x: The derivative of -4x is simply -4, as the power rule applies to x^1, resulting in -4 * 1x^(1-1) = -4.
  3. Differentiating 7: The derivative of a constant is always zero. Therefore, the derivative of 7 is 0.
  4. Differentiating ln(x): The derivative of the natural logarithm function, ln(x), is 1/x. This is a fundamental rule in calculus.

Combining these individual derivatives, we arrive at the first derivative of f(x):

f'(x) = 24x - 4 + 0 + 1/x = 24x - 4 + 1/x

This first derivative, f'(x) = 24x - 4 + 1/x, lays the groundwork for our next step – finding the second derivative.

Step 2: Unveiling the Second Derivative, f''(x)

Now that we have successfully determined the first derivative, f'(x) = 24x - 4 + 1/x, we proceed to the heart of our investigation – finding the second derivative, f''(x). As we know, the second derivative provides insights into the concavity of the function's graph, indicating whether the function is curving upwards or downwards. To find f''(x), we differentiate f'(x) once again, employing the same differentiation rules we used earlier.

Let's meticulously differentiate each term of f'(x) = 24x - 4 + 1/x:

  1. Differentiating 24x: Applying the power rule, the derivative of 24x is simply 24, as 24 * 1x^(1-1) = 24.
  2. Differentiating -4: As the derivative of a constant is zero, the derivative of -4 is 0.
  3. Differentiating 1/x: To differentiate 1/x, we can rewrite it as x^(-1). Applying the power rule, we get -1 * x^(-1-1) = -x^(-2) = -1/x².

Combining these individual derivatives, we arrive at the second derivative of f(x):

f''(x) = 24 - 0 - 1/x² = 24 - 1/x²

Thus, we have successfully unveiled the second derivative of the function f(x) = 12x² - 4x + 7 + ln(x), which is f''(x) = 24 - 1/x². This formula provides us with the key to understanding the concavity of the function's graph at various points.

Step 3: Analyzing the Concavity using f''(x)

With the second derivative, f''(x) = 24 - 1/x², in our grasp, we can now embark on the crucial task of analyzing the concavity of the function f(x) = 12x² - 4x + 7 + ln(x). The sign of f''(x) at a particular point determines the concavity of the function's graph at that point. Let's delve into the analysis:

  1. f''(x) > 0: Concave Up: When the second derivative is positive, f''(x) > 0, the function's graph is concave up, resembling a smile. This means that the rate of change of the slope of the tangent line is increasing, causing the curve to bend upwards.
  2. f''(x) < 0: Concave Down: Conversely, when the second derivative is negative, f''(x) < 0, the function's graph is concave down, resembling a frown. In this case, the rate of change of the slope of the tangent line is decreasing, causing the curve to bend downwards.
  3. f''(x) = 0: Inflection Point (Potential): When the second derivative is zero, f''(x) = 0, we have a potential inflection point. An inflection point is a point on the graph where the concavity changes. However, it's crucial to note that f''(x) = 0 is a necessary but not sufficient condition for an inflection point. We need to further investigate the sign change of f''(x) around that point to confirm whether it is indeed an inflection point.

To illustrate this analysis, let's consider some specific examples:

  • For x > 0, f''(x) = 24 - 1/x² is generally positive because 24 dominates the term 1/x² as x increases. This indicates that the function is concave up for positive values of x.
  • As x approaches 0 from the positive side, 1/x² becomes very large, making f''(x) negative. This suggests that the function might be concave down for values of x very close to 0.
  • To find potential inflection points, we need to solve the equation f''(x) = 0, which translates to 24 - 1/x² = 0. Solving this equation, we get x² = 1/24, which gives us x = ±√(1/24). However, since the natural logarithm function ln(x) is only defined for positive values of x, we only consider the positive root, x = √(1/24).

Further analysis around x = √(1/24) would be required to confirm whether it is indeed an inflection point. We would need to check if the sign of f''(x) changes as we cross this point.

Conclusion

In this comprehensive exploration, we have successfully navigated the realm of derivatives to unveil the second derivative of the function f(x) = 12x² - 4x + 7 + ln(x). Through a meticulous step-by-step process, we first determined the first derivative, f'(x) = 24x - 4 + 1/x, and then proceeded to find the second derivative, f''(x) = 24 - 1/x². This second derivative serves as a powerful tool for analyzing the concavity of the function's graph, allowing us to understand its curvature and identify potential inflection points.

By understanding the relationship between the second derivative and concavity, we gain valuable insights into the behavior of functions. This knowledge is not only fundamental in calculus but also has far-reaching applications in various fields, including physics, engineering, economics, and computer science. The ability to analyze concavity allows us to optimize designs, predict trends, and make informed decisions in a wide range of real-world scenarios. The journey into the world of derivatives and concavity is a testament to the power of calculus in unraveling the complexities of functions and their applications.

Determining the Formula for f''(x)

f(x) = 12x² - 4x + 7 + ln(x): A Step-by-Step Solution for Calculus Students.