Derivative Of Composite Function F(x) = √x And G(x) = 9x³ Explained
If f(x) = √x and g(x) = 9x³, then what is the derivative of the composite function (f o g)(x)?
Introduction
In the realm of calculus, understanding composite functions and their derivatives is crucial for solving complex problems. This article delves into a specific problem involving two functions, f(x) = √x and g(x) = 9x³, aiming to find the derivative of their composite function, (f o g)(x). We'll break down the problem step-by-step, providing a comprehensive explanation to ensure clarity and understanding. This exploration is not just about finding the solution; it's about grasping the underlying concepts and techniques applicable to a wide range of calculus problems. So, let's embark on this mathematical journey, unlocking the secrets of composite functions and derivatives, and empowering ourselves with the tools to tackle similar challenges with confidence. The core concept we'll be focusing on is the chain rule, a fundamental principle in calculus that allows us to differentiate composite functions effectively. The problem presented serves as an excellent example to illustrate the application of the chain rule in a practical scenario. By meticulously dissecting each step, we aim to build a solid foundation for understanding more advanced calculus concepts. Moreover, this article will emphasize the importance of precise notation and methodical problem-solving, two essential skills for any aspiring mathematician or scientist. The journey through this problem will not only enhance your understanding of calculus but also cultivate a deeper appreciation for the elegance and power of mathematical reasoning.
Problem Statement
Given two functions, f(x) = √x and g(x) = 9x³, we aim to determine the derivative of the composite function (f o g)(x). This means we need to find the expression for d/dx [f(g(x))]. The problem at hand requires us to apply the principles of composite functions and the chain rule of differentiation. Understanding these concepts is crucial for tackling the problem effectively. A composite function, in simple terms, is a function that is formed by substituting one function into another. In this case, f(g(x)) means we are plugging the entire function g(x) into the function f(x). The chain rule, on the other hand, is a powerful tool in calculus that allows us to differentiate composite functions. It essentially states that the derivative of a composite function is the product of the derivative of the outer function evaluated at the inner function, and the derivative of the inner function. To successfully solve this problem, we need to first understand how to form the composite function f(g(x)). This involves substituting 9x³ into the √x function. Once we have the composite function, we can then apply the chain rule to find its derivative. This process will involve differentiating both the outer function (f) and the inner function (g) separately and then combining them in the correct manner. The problem provides an excellent opportunity to practice and reinforce our understanding of these fundamental calculus concepts. By carefully following the steps and paying attention to the details, we can arrive at the correct solution and enhance our problem-solving skills in calculus.
Step 1: Forming the Composite Function (f o g)(x)
To begin, we need to find the composite function (f o g)(x), which is f(g(x)). This means we substitute g(x) into f(x). Given f(x) = √x and g(x) = 9x³, we replace the x in f(x) with g(x), which gives us f(g(x)) = √(9x³). This step is fundamental to solving the problem, as it sets the stage for applying the chain rule. Understanding the process of function composition is crucial in calculus and other areas of mathematics. It allows us to combine functions in meaningful ways, creating new functions with potentially complex behaviors. In this specific case, we are essentially taking the square root of the function 9x³. This creates a new function whose behavior is influenced by both the square root function and the cubic function. The composite function f(g(x)) = √(9x³) represents the overall transformation that occurs when we first apply g(x) to an input x, and then apply f(x) to the result. Visualizing this process can be helpful in understanding the concept of function composition. We can think of g(x) as the inner function that performs an initial transformation on x, and f(x) as the outer function that further transforms the output of g(x). This step of forming the composite function is not just a mechanical substitution; it's about understanding the interplay between different functions and how they can be combined to create new mathematical entities. By grasping this concept, we can better appreciate the power and flexibility of functions in modeling real-world phenomena.
Step 2: Simplifying the Composite Function
Before differentiating, it's often beneficial to simplify the composite function. We have f(g(x)) = √(9x³). We can rewrite this as f(g(x)) = √(9) * √(x³), which simplifies to 3√(x³). Further simplification involves expressing the square root as a fractional exponent: 3(x³)^(1/2). Applying the power of a power rule, we get 3x^(3/2). Simplifying the function before differentiation is a crucial step in calculus. It not only makes the differentiation process easier but also reduces the chances of making errors. In this case, we have simplified √(9x³) to 3x^(3/2), which is a much more manageable form for differentiation. The simplification process involves using various algebraic rules and properties. We first separated the square root of 9 from the square root of x³, which allowed us to simplify √(9) to 3. Then, we expressed the square root of x³ as x^(3/2) using the relationship between radicals and fractional exponents. This step demonstrates the importance of having a strong foundation in algebra when working with calculus problems. The ability to manipulate algebraic expressions efficiently can significantly simplify complex calculations and lead to a clearer understanding of the underlying mathematical relationships. By simplifying the composite function, we have transformed it into a form that is readily amenable to differentiation using the power rule, a fundamental rule in calculus. This highlights the strategic nature of problem-solving in mathematics. Often, a problem can be made much easier by choosing the right approach and applying the appropriate simplification techniques. This step not only simplifies the calculation but also provides a deeper insight into the structure of the composite function, making it easier to visualize and understand its behavior.
Step 3: Applying the Chain Rule
Now, we differentiate 3x^(3/2) with respect to x. The power rule states that d/dx(x^n) = nx^(n-1). Applying this rule, we get d/dx [3x^(3/2)] = 3 * (3/2) * x^((3/2) - 1) = (9/2)x^(1/2). Thus, the derivative of (f o g)(x) is (9/2)√x. This is the core of the solution, where we utilize the fundamental concept of the chain rule (though in this simplified form, it primarily involves the power rule). The chain rule, in its general form, is used to differentiate composite functions of the form f(g(x)). However, in this specific case, after simplification, we were able to directly apply the power rule, which is a special case of the chain rule. The power rule is a cornerstone of differential calculus, providing a straightforward method for differentiating functions of the form x^n, where n is a constant. Applying the power rule involves multiplying the function by the exponent n and then reducing the exponent by 1. In our case, we had the function 3x^(3/2). Applying the power rule, we multiplied the coefficient 3 by the exponent 3/2, resulting in 9/2. Then, we reduced the exponent by 1, from 3/2 to 1/2. This gave us the derivative (9/2)x^(1/2). The final step involves expressing x^(1/2) as √x, which is a common and often preferred notation. This leads us to the final answer: (9/2)√x. This step not only demonstrates the application of the power rule but also highlights the importance of mathematical notation and the ability to move seamlessly between different representations of the same mathematical concept. The derivative (9/2)√x represents the instantaneous rate of change of the composite function f(g(x)) with respect to x. It provides valuable information about the function's behavior, such as its increasing or decreasing nature and its concavity.
Step 4: Expressing the Result
The derivative of (f o g)(x) is (9/2)√x. This is the final answer, representing the rate of change of the composite function with respect to x. We can also write this as 4.5√x. The final result, (9/2)√x, encapsulates the entire process of forming the composite function, simplifying it, and then differentiating it using the power rule. This derivative represents the instantaneous rate of change of the composite function f(g(x)) = 3x^(3/2) with respect to the variable x. In simpler terms, it tells us how much the output of the composite function changes for a small change in the input x. The expression (9/2)√x is a concise and mathematically precise way to represent this rate of change. However, it can also be expressed in different forms, such as 4.5√x, which is simply a decimal representation of the coefficient 9/2. The choice of representation often depends on the context and the desired level of precision. Understanding the interpretation of the derivative is crucial in calculus and its applications. The derivative is not just a mathematical formula; it has a tangible meaning in terms of rates of change and slopes of tangent lines. In this case, the derivative (9/2)√x tells us that the rate of change of the composite function increases as x increases. This is because the square root function √x is an increasing function, and the coefficient 9/2 scales this increase. Expressing the result clearly and concisely is an important aspect of mathematical communication. It ensures that the solution is easily understood and can be readily used in further calculations or analysis. This final step emphasizes the importance of not just finding the answer but also presenting it in a meaningful and interpretable way.
Conclusion
In summary, given f(x) = √x and g(x) = 9x³, the derivative of (f o g)(x) is (9/2)√x or 4.5√x. This solution involved forming the composite function, simplifying it using algebraic rules, and then applying the power rule of differentiation. Understanding composite functions and the chain rule is essential in calculus. This problem provided a practical application of these concepts. The process of solving this problem highlights the importance of a step-by-step approach in mathematics. We started by clearly defining the problem, then systematically worked through each step, from forming the composite function to simplifying it and finally differentiating it. This methodical approach is crucial for tackling complex problems and minimizing errors. The use of algebraic simplification played a significant role in making the differentiation process easier. By simplifying the composite function before differentiating, we were able to apply the power rule directly, avoiding the need for a more complex application of the chain rule. This demonstrates the value of having a strong foundation in algebra and the ability to manipulate expressions effectively. The application of the power rule is a fundamental skill in differential calculus. This problem provided an opportunity to reinforce this skill and understand its application in the context of composite functions. The final result, (9/2)√x, represents the rate of change of the composite function and provides valuable information about its behavior. This underscores the importance of understanding the interpretation of derivatives in calculus. Overall, this problem serves as a valuable exercise in applying key calculus concepts and techniques. By carefully working through each step and understanding the underlying principles, we can enhance our problem-solving skills and gain a deeper appreciation for the power and elegance of calculus. The ability to solve problems like this is essential for further study in mathematics, science, and engineering.