Derivative Of (x^2 + 4x + 2)^4 A Step-by-Step Solution
In the realm of calculus, the derivative of a function, often denoted as f'(x), provides invaluable insights into the rate at which the function's output changes with respect to its input. This concept is fundamental in numerous fields, including physics, engineering, economics, and computer science, where understanding rates of change is paramount. In this comprehensive exploration, we will delve into the intricacies of finding the derivative of a composite function, specifically focusing on the example f(x) = (x^2 + 4x + 2)^4. This function exemplifies a scenario where the chain rule, a cornerstone of differential calculus, comes into play. We will dissect the process of applying the chain rule, providing a step-by-step guide to calculating f'(x). Furthermore, we will evaluate f'(1), demonstrating how to substitute a specific value into the derived function to obtain the instantaneous rate of change at that point. This exploration aims to provide a thorough understanding of composite functions and their derivatives, equipping readers with the skills to tackle similar problems with confidence.
Applying the Chain Rule: A Step-by-Step Guide
When faced with a composite function like f(x) = (x^2 + 4x + 2)^4, the chain rule is our indispensable tool. The chain rule elegantly addresses the differentiation of composite functions, which are functions formed by nesting one function inside another. In essence, it 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 itself. To effectively apply the chain rule, we must first identify the outer and inner functions. In our case, the outer function is g(u) = u^4, where u represents the inner function. The inner function is h(x) = x^2 + 4x + 2. The chain rule then dictates that the derivative of f(x) is given by f'(x) = g'(h(x)) * h'(x). This formula may seem daunting at first, but by systematically breaking it down, we can navigate it with ease. We begin by finding the derivatives of the outer and inner functions separately. The derivative of g(u) = u^4 is g'(u) = 4u^3, using the power rule of differentiation. The derivative of h(x) = x^2 + 4x + 2 is h'(x) = 2x + 4, applying the power rule and the sum rule of differentiation. Next, we substitute the inner function h(x) into g'(u), obtaining g'(h(x)) = 4(x^2 + 4x + 2)^3. Finally, we multiply g'(h(x)) by h'(x) to arrive at the derivative of f(x): f'(x) = 4(x^2 + 4x + 2)^3 * (2x + 4). This methodical application of the chain rule allows us to successfully differentiate complex composite functions.
Calculating the Derivative f'(x)
Having established the framework of the chain rule, let's meticulously calculate the derivative of our function, f(x) = (x^2 + 4x + 2)^4. As we dissected earlier, the chain rule guides us to differentiate the outer function while keeping the inner function intact, then multiply by the derivative of the inner function. Following this principle, we first focus on the outer function, which is the power function raised to the fourth power. Applying the power rule, we bring down the exponent (4) and reduce the exponent by one, resulting in 4(x^2 + 4x + 2)^3. Notice that we have kept the inner function, x^2 + 4x + 2, untouched within the parentheses. This is a crucial step in the chain rule process. Next, we turn our attention to the inner function, h(x) = x^2 + 4x + 2. We differentiate this function with respect to x. The derivative of x^2 is 2x, the derivative of 4x is 4, and the derivative of the constant 2 is 0. Thus, the derivative of the inner function is h'(x) = 2x + 4. Now, we multiply the derivative of the outer function, 4(x^2 + 4x + 2)^3, by the derivative of the inner function, 2x + 4. This yields the complete derivative: f'(x) = 4(x^2 + 4x + 2)^3 * (2x + 4). This expression represents the instantaneous rate of change of the function f(x) at any given point x. To further refine our result, we can factor out a 2 from the (2x + 4) term, giving us f'(x) = 8(x^2 + 4x + 2)^3 * (x + 2). This simplified form is often preferred for its conciseness and ease of use in subsequent calculations.
Evaluating f'(1): Finding the Instantaneous Rate of Change
Now that we have successfully determined the derivative function, f'(x) = 8(x^2 + 4x + 2)^3 * (x + 2), the next logical step is to evaluate this derivative at a specific point. In this case, we are interested in finding f'(1), which represents the instantaneous rate of change of the function f(x) at x = 1. To find f'(1), we simply substitute x = 1 into the expression for f'(x). This means replacing every instance of x in the equation with the value 1. So, we have f'(1) = 8((1)^2 + 4(1) + 2)^3 * (1 + 2). Now, we perform the arithmetic operations within the parentheses. (1)^2 is 1, 4(1) is 4, so the inner expression becomes (1 + 4 + 2), which simplifies to 7. Therefore, we have f'(1) = 8(7)^3 * (1 + 2). Next, we calculate 7 cubed, which is 7 * 7 * 7 = 343. And (1 + 2) is simply 3. So, our expression now reads f'(1) = 8 * 343 * 3. Finally, we multiply these values together. 8 multiplied by 343 is 2744, and 2744 multiplied by 3 is 8232. Thus, we arrive at the result: f'(1) = 8232. This value signifies that at the point x = 1, the function f(x) is changing at a rate of 8232 units for every one unit change in x. This result provides a concrete understanding of the function's behavior at a specific point, highlighting the power of the derivative in analyzing rates of change.
Connecting the Derivative to Tangent Lines
The derivative, f'(x), is not merely an abstract mathematical concept; it has a profound geometric interpretation. The value of the derivative at a specific point, such as f'(1) = 8232, represents the slope of the tangent line to the graph of the function f(x) at that point. Imagine the graph of f(x) as a curve in the coordinate plane. At the point where x = 1, we can draw a straight line that touches the curve at that point and has the same instantaneous direction as the curve. This line is the tangent line. The slope of this tangent line is precisely the value of the derivative at x = 1, which we calculated to be 8232. A steep positive slope, like 8232, indicates that the function is increasing rapidly at that point. Conversely, a negative slope would indicate that the function is decreasing, and a slope of zero would indicate a horizontal tangent, where the function has a local maximum or minimum. This connection between the derivative and the tangent line provides a visual and intuitive understanding of the derivative's meaning. It allows us to visualize how the function is changing at a particular point and to relate this change to the geometric properties of the function's graph. Furthermore, the tangent line provides a linear approximation of the function near the point of tangency. This approximation is a powerful tool in various applications, allowing us to estimate the function's value at nearby points using a simple linear equation.
Applications of Derivatives in Real-World Scenarios
The concept of derivatives extends far beyond the realm of theoretical mathematics and finds widespread applications in various real-world scenarios. Understanding derivatives allows us to model and analyze rates of change in diverse fields, providing valuable insights and predictive capabilities. In physics, derivatives are fundamental in describing motion. The derivative of an object's position with respect to time gives its velocity, and the derivative of velocity with respect to time gives its acceleration. These concepts are essential for understanding and predicting the motion of objects, from projectiles to planets. In engineering, derivatives are used to optimize designs and processes. For example, engineers might use derivatives to determine the optimal shape of an airplane wing to minimize drag or to design a bridge that can withstand maximum stress. In economics, derivatives are used to analyze marginal cost and marginal revenue, helping businesses make informed decisions about production and pricing. The concept of elasticity, which measures the responsiveness of demand or supply to changes in price or income, is also based on derivatives. In computer science, derivatives are used in machine learning algorithms, particularly in optimization techniques like gradient descent, which are used to train models to minimize errors. In finance, derivatives are used to model and manage risk, and to price financial instruments such as options and futures. These are just a few examples of the many applications of derivatives in real-world scenarios. The ability to understand and apply derivatives is a valuable skill in a wide range of disciplines.
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