Determining G'(z) From First Principles And Analyzing Function Behavior At Z=-2
In this comprehensive analysis, we embark on a journey to determine the derivative, g'(z), of the function g(z) = -5 / (2z^2 + 6) using the fundamental first principles approach. Understanding this method is crucial as it provides a robust foundation for grasping the core concept of differentiation. We will then leverage this derivative to meticulously assess the function's behavior—whether it is increasing, decreasing, or exhibiting neither trend—specifically at the point z = -2. This exploration is not just a mathematical exercise; it is a deep dive into the dynamics of functions and their rates of change, equipping us with powerful analytical tools.
The concept of derivatives is central to calculus and mathematical analysis. Derivatives describe the instantaneous rate of change of a function, a critical understanding for various applications across science, engineering, and economics. The first principles method, also known as the delta method or the limit definition of the derivative, allows us to compute derivatives directly from the definition, circumventing the need for pre-established rules or formulas. This approach is particularly valuable for educational purposes, as it vividly illustrates the underlying mechanics of differentiation.
Applying First Principles to Find g'(z)
To kick off our analysis, we recall the definition of the derivative from first principles:
g'(z) = lim (h->0) [g(z + h) - g(z)] / h
This formula calculates the derivative g'(z) by evaluating the limit of the difference quotient as h approaches zero. The difference quotient represents the average rate of change of the function over a small interval, and the limit captures the instantaneous rate of change at a specific point. Substituting our function g(z) = -5 / (2z^2 + 6) into this definition, we get:
g'(z) = lim (h->0) [-5 / (2(z + h)^2 + 6) - (-5 / (2z^2 + 6))] / h
This expression forms the heart of our calculation. Our next step involves simplifying this complex fraction to make it amenable to evaluating the limit. We begin by expanding the term (z + h)^2, which gives us z^2 + 2zh + h^2. Substituting this back into our expression, we have:
g'(z) = lim (h->0) [-5 / (2(z^2 + 2zh + h^2) + 6) + 5 / (2z^2 + 6)] / h
Next, we distribute the 2 in the denominator, simplifying the expression further:
g'(z) = lim (h->0) [-5 / (2z^2 + 4zh + 2h^2 + 6) + 5 / (2z^2 + 6)] / h
To combine the fractions in the numerator, we find a common denominator. The common denominator is the product of the two denominators, which is (2z^2 + 4zh + 2h^2 + 6)(2z^2 + 6). We rewrite the fractions with this common denominator:
g'(z) = lim (h->0) [(-5(2z^2 + 6) + 5(2z^2 + 4zh + 2h^2 + 6)) / ((2z^2 + 4zh + 2h^2 + 6)(2z^2 + 6))] / h
Now, we expand and simplify the numerator:
g'(z) = lim (h->0) [(-10z^2 - 30 + 10z^2 + 20zh + 10h^2 + 30) / ((2z^2 + 4zh + 2h^2 + 6)(2z^2 + 6))] / h
Notice that the -10z^2 and +10z^2 terms cancel out, as do the -30 and +30 terms, leaving us with:
g'(z) = lim (h->0) [(20zh + 10h^2) / ((2z^2 + 4zh + 2h^2 + 6)(2z^2 + 6))] / h
We can factor out an h from the numerator:
g'(z) = lim (h->0) [h(20z + 10h) / ((2z^2 + 4zh + 2h^2 + 6)(2z^2 + 6))] / h
The h in the numerator and the h in the denominator cancel out, simplifying the expression to:
g'(z) = lim (h->0) (20z + 10h) / ((2z^2 + 4zh + 2h^2 + 6)(2z^2 + 6))
Finally, we take the limit as h approaches 0. As h tends to 0, the terms involving h vanish:
g'(z) = (20z + 10(0)) / ((2z^2 + 4z(0) + 2(0)^2 + 6)(2z^2 + 6))
g'(z) = 20z / ((2z^2 + 6)(2z^2 + 6))
This simplifies to:
g'(z) = 20z / (2z^2 + 6)^2
Thus, we have successfully determined the derivative of g(z) from first principles. This result, g'(z) = 20z / (2z^2 + 6)^2, will be pivotal in our subsequent analysis of the function's behavior at z = -2.
Analyzing Function Behavior at z = -2
Having computed the derivative g'(z) = 20z / (2z^2 + 6)^2 using the first principles method, we now shift our focus to analyzing the behavior of the function g(z) at the specific point z = -2. Our primary goal is to ascertain whether the function is increasing, decreasing, or exhibiting a stationary trend at this point. The derivative, g'(z), serves as a crucial indicator of this behavior; its sign at a particular point directly corresponds to the function's trend.
The sign of the derivative at a point provides valuable insights into the function's behavior. If g'(z) > 0 at a point, it signifies that the function is increasing at that point. Conversely, if g'(z) < 0, the function is decreasing. If g'(z) = 0, the function has a stationary point, which could be a local maximum, a local minimum, or an inflection point. Therefore, by evaluating g'(-2), we can determine the function's trend at z = -2.
Evaluating g'(-2)
To determine the function's behavior at z = -2, we substitute z = -2 into our derivative expression:
g'(-2) = 20(-2) / (2(-2)^2 + 6)^2
First, we calculate (-2)^2, which equals 4. Substituting this back into the expression, we get:
g'(-2) = -40 / (2(4) + 6)^2
Next, we perform the multiplication and addition inside the parentheses:
g'(-2) = -40 / (8 + 6)^2
g'(-2) = -40 / (14)^2
Now, we calculate 14^2, which equals 196:
g'(-2) = -40 / 196
This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 4:
g'(-2) = -10 / 49
Interpreting the Result
We have found that g'(-2) = -10 / 49. The crucial observation here is that g'(-2) is a negative value. As discussed earlier, a negative derivative indicates that the function is decreasing at that point. Therefore, we can confidently conclude that the function g(z) is decreasing at z = -2.
The negative value of g'(-2) provides a clear and definitive answer to our question about the function's behavior at z = -2. This analysis underscores the power of the derivative as a tool for understanding the dynamics of functions. By simply evaluating the sign of the derivative, we have gained valuable insight into the function's trend at a specific point.
Summary of Findings
In summary, we embarked on a detailed exploration of the function g(z) = -5 / (2z^2 + 6). Initially, we employed the first principles method to derive the derivative, successfully obtaining g'(z) = 20z / (2z^2 + 6)^2. This process not only reinforced our understanding of the fundamental definition of the derivative but also provided us with a crucial tool for subsequent analysis.
Following the derivation, we directed our attention to analyzing the function's behavior at z = -2. By substituting z = -2 into the derivative, we calculated g'(-2) = -10 / 49. The negative sign of this value unequivocally indicated that the function g(z) is decreasing at z = -2. This conclusion was a direct application of the principle that a negative derivative signifies a decreasing function.
This comprehensive analysis highlights the interconnectedness of mathematical concepts. The first principles method, the derivative, and the function's behavior are all intimately linked. By mastering these concepts and their applications, we equip ourselves with a robust toolkit for tackling a wide array of mathematical challenges. The ability to derive derivatives from first principles and interpret their values is a cornerstone of calculus and mathematical analysis, enabling us to understand and predict the behavior of functions across various contexts.
This detailed exploration serves not only as a solution to the specific problem but also as a demonstration of a broader analytical approach. The combination of theoretical understanding and practical application is key to mathematical proficiency. By meticulously working through each step, we have not only arrived at the correct answer but also deepened our understanding of the underlying principles and techniques involved. This holistic approach is essential for success in mathematics and related fields.
In conclusion, through the rigorous application of first principles, we have successfully determined the derivative of the function g(z) = -5 / (2z^2 + 6) to be g'(z) = 20z / (2z^2 + 6)^2. Furthermore, by evaluating this derivative at z = -2, we have definitively established that the function is decreasing at this point. This comprehensive analysis underscores the power of calculus in understanding the behavior of functions and provides a solid foundation for further mathematical explorations.
