Finding The Second Derivative Of G(x) = 6x³eˣ A Step-by-Step Solution

Finding the second derivative of a function can seem daunting, but with a systematic approach and a solid understanding of calculus principles, it becomes a manageable task. In this comprehensive guide, we will meticulously walk through the process of finding the second derivative of the function g(x) = 6x³eˣ. This exploration will not only provide the solution but also illuminate the underlying concepts and techniques involved in differentiation. Derivatives are a cornerstone of calculus, playing a pivotal role in understanding rates of change, optimization problems, and the behavior of functions. Mastering the art of differentiation, especially finding higher-order derivatives, is crucial for anyone delving deeper into mathematical analysis and its applications in various fields such as physics, engineering, and economics. We will leverage the product rule, a fundamental concept in differential calculus, to navigate the complexities arising from the function's structure. The product rule states that the derivative of the product of two functions is the derivative of the first function multiplied by the second function, plus the first function multiplied by the derivative of the second function. This rule is indispensable when dealing with functions that are products of other functions, like our g(x). The exponential function, eˣ, holds a special place in calculus due to its unique property of being its own derivative. This simplifies calculations and makes it a recurring element in many mathematical models. Understanding its behavior and how it interacts with other functions is key to tackling a wide array of problems. The power rule, another essential tool in our calculus arsenal, allows us to differentiate polynomial terms efficiently. It states that the derivative of xⁿ is nxⁿ⁻¹, where n is any real number. This rule is particularly useful for handling the x³ term in our function. By combining these rules and applying them meticulously, we will unravel the intricacies of g(x) and arrive at its second derivative. The journey involves finding the first derivative, simplifying the result, and then differentiating again to obtain the second derivative. Each step requires careful attention to detail and a firm grasp of the underlying principles. This process is not just about finding an answer; it's about honing our mathematical skills and developing a deeper appreciation for the elegance and power of calculus. Through this exploration, we aim to not only provide the solution but also equip you with the knowledge and confidence to tackle similar problems independently. Let's embark on this mathematical adventure and unlock the secrets of the second derivative of g(x) = 6x³eˣ.

Step-by-Step Differentiation of g(x) = 6x³eˣ

To find the second derivative, g''(x), we must first determine the first derivative, g'(x). Given g(x) = 6x³eˣ, we will employ the product rule, which is essential when dealing with the product of two functions. The product rule states that if we have two functions, u(x) and v(x), then the derivative of their product is given by (uv)' = u'v + uv'. In our case, we can identify u(x) = 6x³ and v(x) = eˣ. Differentiating u(x) with respect to x, we get u'(x) = 18x². This follows from the power rule, which states that the derivative of xⁿ is nxⁿ⁻¹. The power rule is a fundamental concept in differential calculus and is widely used to find the derivatives of polynomial functions. Differentiating v(x) = eˣ with respect to x, we find that v'(x) = eˣ. The exponential function eˣ is unique in that its derivative is itself, making it a recurring element in many mathematical models and applications. Now, applying the product rule, we have:

g'(x) = u'(x)v(x) + u(x)v'(x) g'(x) = (18x²)(eˣ) + (6x³)(eˣ)

This expression represents the first derivative of g(x). To find the second derivative, we must differentiate g'(x) with respect to x. This involves differentiating the sum of two terms, each of which is a product of functions. Therefore, we will need to apply the product rule again, this time to each term in g'(x). The ability to apply differentiation rules repeatedly and in combination is a crucial skill in calculus. It allows us to tackle complex functions and unravel their properties. Let's proceed by differentiating each term separately.

For the first term, 18x²eˣ, we identify u(x) = 18x² and v(x) = eˣ. Differentiating u(x), we get u'(x) = 36x, again using the power rule. Differentiating v(x), we still have v'(x) = eˣ, as the derivative of eˣ remains eˣ. Applying the product rule to this term, we get:

(18x²eˣ)' = (36x)(eˣ) + (18x²)(eˣ)

For the second term, 6x³eˣ, we have u(x) = 6x³ and v(x) = eˣ. Differentiating u(x), we find u'(x) = 18x², once again using the power rule. The derivative of v(x) remains v'(x) = eˣ. Applying the product rule to this term, we get:

(6x³eˣ)' = (18x²)(eˣ) + (6x³)(eˣ)

Now, we combine the derivatives of both terms to find the second derivative g''(x):

g''(x) = (36x)(eˣ) + (18x²)(eˣ) + (18x²)(eˣ) + (6x³)(eˣ)

This expression gives us the second derivative of g(x). However, it can be simplified to a more compact form, which will make it easier to work with in further calculations or analysis.

Simplifying the Second Derivative

After applying the product rule twice, we've arrived at the expression for the second derivative, g''(x) = (36x)(eˣ) + (18x²)(eˣ) + (18x²)(eˣ) + (6x³)(eˣ). This expression, while mathematically correct, is not in its simplest form. Simplifying the expression not only makes it more aesthetically pleasing but also enhances its usability in subsequent calculations or analyses. Simplification often involves combining like terms and factoring out common factors. In this case, we notice that all terms contain the exponential function eˣ, making it a prime candidate for factoring. Factoring is a powerful technique in algebra and calculus that allows us to rewrite expressions in a more concise and manageable form. It involves identifying common factors among terms and extracting them, thereby reducing the complexity of the expression. Moreover, we observe that there are two terms with x²eˣ, which can be combined. Combining like terms is a fundamental algebraic operation that involves adding or subtracting terms that have the same variable raised to the same power. This simplifies the expression by reducing the number of terms and making it easier to understand. By combining these terms, we can further streamline the expression and make it more amenable to analysis. Let's start by combining the like terms:

g''(x) = (36x)(eˣ) + (18x² + 18x²)(eˣ) + (6x³)(eˣ) g''(x) = (36x)(eˣ) + (36x²)(eˣ) + (6x³)(eˣ)

Now, we can factor out the common factor of 6xeˣ from each term. Factoring out common factors is a crucial step in simplifying algebraic expressions. It allows us to rewrite the expression in a more compact form, which can be particularly useful when dealing with complex equations or functions. By factoring out the common factor, we can reveal the underlying structure of the expression and make it easier to analyze. Factoring also helps in identifying potential roots or zeros of the function, which are critical in many mathematical applications. By extracting 6xeˣ, we are essentially isolating the core components of the expression, making it easier to interpret and manipulate. This step not only simplifies the expression but also provides insights into its behavior and properties.

g''(x) = 6xeˣ(6 + 6x + x²)

This is the simplified form of the second derivative. The expression 6xeˣ(6 + 6x + x²) is much more compact and easier to work with compared to the original expanded form. The quadratic expression within the parentheses, 6 + 6x + x², can be further analyzed to determine its roots and behavior. Understanding the roots of a function is essential in various mathematical applications, including optimization problems, curve sketching, and stability analysis. The roots of a function are the values of the variable that make the function equal to zero. These values provide critical information about the function's behavior, such as where it crosses the x-axis and where it changes its sign. Analyzing the quadratic expression allows us to gain a deeper understanding of the second derivative and its implications for the original function, g(x). The factored form also provides insights into the points where the second derivative might be zero, which are potential inflection points of the original function. Inflection points are points where the concavity of a curve changes, and they play a crucial role in understanding the shape and behavior of the function. By simplifying the second derivative, we have not only made it more manageable but also unlocked valuable information about the function's properties and behavior. This simplified form is now ready to be used in further analysis, such as finding inflection points or determining the concavity of the function g(x). In the next section, we will discuss the significance of the second derivative and how it relates to the original function.

Significance of the Second Derivative

The second derivative, g''(x), plays a pivotal role in understanding the behavior of the original function, g(x). It provides insights into the concavity of the function, which is a fundamental aspect of its shape and curvature. Concavity refers to the direction in which a curve bends. A function is said to be concave up if its graph curves upwards, like a smile, and concave down if its graph curves downwards, like a frown. The second derivative acts as a concavity detector, telling us whether the function is curving upwards or downwards at a given point. This information is invaluable in various applications, such as optimization problems, where we seek to find maximum or minimum values, and curve sketching, where we aim to create an accurate representation of the function's graph. The sign of the second derivative determines the concavity: if g''(x) > 0, the function is concave up; if g''(x) < 0, the function is concave down; and if g''(x) = 0, we have a potential inflection point. Understanding the relationship between the second derivative and concavity is crucial for analyzing the behavior of functions and making informed decisions in various mathematical and real-world scenarios. Inflection points, where the concavity changes, are particularly significant features of a function. They mark the transition from concave up to concave down or vice versa, and they often correspond to significant changes in the function's behavior. Inflection points can be located by finding the points where the second derivative is equal to zero or undefined. These points provide valuable information about the function's shape and can help us understand its overall behavior. In addition to determining concavity and locating inflection points, the second derivative is also used in the second derivative test, a powerful tool for finding local maxima and minima of a function. The second derivative test complements the first derivative test, providing additional information about the critical points of the function. By examining the sign of the second derivative at a critical point, we can determine whether the point is a local maximum (where the function reaches a peak), a local minimum (where the function reaches a valley), or neither. The second derivative test is particularly useful when the first derivative test is inconclusive or difficult to apply. It provides a straightforward method for classifying critical points and understanding the function's behavior in their vicinity. Moreover, the second derivative finds applications in various fields beyond mathematics. In physics, it is related to acceleration, which is the rate of change of velocity. In economics, it can be used to analyze the rate of change of marginal cost or marginal revenue. In engineering, it plays a role in understanding the stability of structures and systems. The versatility of the second derivative makes it a fundamental concept in many scientific and engineering disciplines. Understanding its properties and applications is essential for anyone working with mathematical models and simulations. In summary, the second derivative is a powerful tool for analyzing the behavior of functions. It provides insights into concavity, helps locate inflection points, and aids in finding local maxima and minima. Its applications extend beyond mathematics into various fields, making it a cornerstone of scientific and engineering analysis. By mastering the concept of the second derivative, we equip ourselves with a valuable tool for understanding and interpreting the world around us.

Conclusion

In conclusion, finding the second derivative of g(x) = 6x³eˣ involved a meticulous application of the product rule, followed by simplification techniques. The final simplified form of the second derivative, g''(x) = 6xeˣ(x² + 6x + 6), provides valuable insights into the behavior of the original function. This journey through differentiation highlights the importance of mastering fundamental calculus concepts and techniques. Calculus, with its tools for analyzing change and motion, forms the backbone of many scientific and engineering disciplines. The ability to differentiate functions, especially finding higher-order derivatives, is crucial for understanding rates of change, optimization problems, and the behavior of complex systems. The product rule, which we employed extensively in this exploration, is a cornerstone of differential calculus. It allows us to differentiate functions that are products of other functions, a common occurrence in mathematical modeling. The exponential function, eˣ, with its unique property of being its own derivative, plays a significant role in various mathematical and scientific contexts. Understanding its behavior and how it interacts with other functions is essential for tackling a wide array of problems. The power rule, another fundamental tool in our calculus arsenal, allows us to differentiate polynomial terms efficiently. The power rule is widely used in calculus and its applications, providing a simple and effective way to find the derivatives of polynomial functions. By combining these rules and applying them systematically, we can navigate the complexities of differentiation and arrive at accurate solutions. Simplification techniques, such as combining like terms and factoring out common factors, are essential for making expressions more manageable and interpretable. Simplified expressions are not only easier to work with but also provide clearer insights into the underlying structure and behavior of functions. The second derivative, g''(x), provides crucial information about the concavity of the original function, g(x). Understanding concavity is essential for analyzing the shape and behavior of functions, identifying inflection points, and solving optimization problems. The second derivative test, which utilizes the sign of the second derivative to classify critical points, is a powerful tool for finding local maxima and minima of a function. This test complements the first derivative test, providing a comprehensive approach to identifying and classifying critical points. The concepts and techniques explored in this guide have broad applications in various fields, including physics, engineering, economics, and computer science. Calculus is a fundamental tool in these disciplines, providing the mathematical framework for modeling and analyzing complex systems. By mastering the art of differentiation and understanding the significance of higher-order derivatives, we equip ourselves with the skills and knowledge necessary to tackle a wide range of mathematical and scientific challenges. This exploration serves as a testament to the power and elegance of calculus, demonstrating its ability to unravel the intricacies of functions and provide valuable insights into their behavior. We hope that this comprehensive guide has not only provided the solution to the given problem but also deepened your understanding of calculus principles and their applications. Keep practicing, keep exploring, and keep unlocking the power of mathematics!