Differentiating Y = 4x^-5 - 2x^-1 A Step-by-Step Guide
Introduction
In calculus, differentiation is a fundamental operation that allows us to find the rate of change of a function. Specifically, it helps us determine the derivative of a function, which represents the slope of the tangent line at any given point on the function's graph. This article focuses on differentiating the function y = 4x^{-5} - 2x^{-1}. We will walk through the process step-by-step, utilizing the power rule and other essential differentiation techniques to arrive at the solution. This exploration is crucial not only for mastering calculus but also for understanding various real-world applications where rates of change are significant.
Understanding the Power Rule
The power rule is a cornerstone of differentiation, providing a straightforward method for finding the derivative of power functions. A power function is any function of the form f(x) = ax^n, where a is a constant and n is any real number. According to the power rule, the derivative of f(x) is given by f'(x) = nax^{n-1}. In simpler terms, we multiply the coefficient a by the exponent n and then reduce the exponent by 1. This rule is essential for differentiating polynomials and other functions involving powers of x. When dealing with more complex functions, we often apply the power rule in conjunction with other differentiation techniques such as the sum/difference rule, the product rule, and the chain rule. The power rule's versatility makes it an indispensable tool in calculus, enabling us to tackle a wide range of differentiation problems efficiently. Understanding and mastering the power rule is crucial for anyone delving into calculus and its applications in various fields.
Applying the Sum/Difference Rule
The sum/difference rule is another essential concept in calculus that simplifies the differentiation of functions involving sums or differences of terms. This rule states that the derivative of a sum or difference of functions is the sum or difference of their derivatives. Mathematically, if we have a function y = u(x) ± v(x), where u(x) and v(x) are differentiable functions, then the derivative dy/dx is given by dy/dx = u'(x) ± v'(x). This rule allows us to differentiate complex functions by breaking them down into simpler components and differentiating each term separately. For example, if we have a function like y = 3x^2 + 2x - 1, we can differentiate each term individually: the derivative of 3x^2 is 6x, the derivative of 2x is 2, and the derivative of -1 is 0. Thus, the derivative of the entire function is 6x + 2. The sum/difference rule is particularly useful when combined with other differentiation rules, such as the power rule, making it a fundamental tool in calculus. It enables us to tackle more complex problems systematically and efficiently, enhancing our understanding of function behavior and rates of change.
Step-by-Step Differentiation of y = 4x^{-5} - 2x^{-1}
To differentiate the function y = 4x^{-5} - 2x^{-1}, we will apply the power rule and the sum/difference rule. Here’s a detailed breakdown:
- Identify the terms: The function consists of two terms: 4x^{-5} and -2x^{-1}.
- Apply the power rule to the first term: For 4x^{-5}, the power rule states that we multiply the coefficient (4) by the exponent (-5) and reduce the exponent by 1. So, the derivative of 4x^{-5} is 4 * (-5) * x^{-5-1} = -20x^{-6}.
- Apply the power rule to the second term: For -2x^{-1}, we multiply the coefficient (-2) by the exponent (-1) and reduce the exponent by 1. Thus, the derivative of -2x^{-1} is -2 * (-1) * x^{-1-1} = 2x^{-2}.
- Combine the derivatives: Using the sum/difference rule, we add the derivatives of each term to find the derivative of the entire function. So, dy/dx = -20x^{-6} + 2x^{-2}.
Thus, the derivative of y = 4x^{-5} - 2x^{-1} is dy/dx = -20x^{-6} + 2x^{-2}. This step-by-step approach ensures a clear understanding of the differentiation process, making it easier to tackle more complex problems.
Detailed Application of the Power Rule
The power rule is a fundamental concept in calculus, essential for finding the derivatives of functions in the form f(x) = ax^n, where a is a constant coefficient and n is the exponent. The rule states that the derivative f'(x) is given by nax^{n-1}. This means we multiply the coefficient by the exponent and then reduce the exponent by one. To illustrate, let’s apply this rule to our function, y = 4x^{-5} - 2x^{-1}. The function consists of two terms, each of which we will differentiate separately using the power rule.
For the first term, 4x^-5}*, the coefficient a is 4 and the exponent n is -5. Applying the power rule, we multiply the coefficient by the exponent is -20x^-6}*. This process transforms the original power function into its derivative, showing how the function’s rate of change varies with x. Similarly, for the second term, -2x^{-1}, the coefficient a is -2 and the exponent n is -1. Applying the power rule, we multiply the coefficient by the exponent is 2x^{-2}. This step-by-step breakdown highlights the mechanical application of the power rule, making it easier to understand how each part of the term contributes to the final derivative. Mastering the power rule is crucial, as it is frequently used in various calculus problems and forms the basis for differentiating more complex functions.
Comprehensive Use of the Sum/Difference Rule
The sum/difference rule is a critical tool in calculus that simplifies the differentiation of functions composed of multiple terms added or subtracted together. This rule states that the derivative of a sum or difference of functions is the sum or difference of their individual derivatives. Mathematically, if we have a function y = u(x) ± v(x), where u(x) and v(x) are differentiable functions, then the derivative dy/dx is given by dy/dx = u'(x) ± v'(x). In our specific case, the function y = 4x^{-5} - 2x^{-1} consists of two terms, 4x^{-5} and -2x^{-1}, which are subtracted. To apply the sum/difference rule, we first differentiate each term separately and then combine the results.
We've already found the derivatives of each term using the power rule. The derivative of 4x^{-5} is -20x^{-6}, and the derivative of -2x^{-1} is 2x^{-2}. Now, applying the sum/difference rule, we add these derivatives together to find the derivative of the entire function. Thus, the derivative dy/dx is given by -20x^{-6} + 2x^{-2}. This straightforward addition combines the rates of change of each term, providing a comprehensive view of how the entire function changes. The sum/difference rule allows us to break down complex differentiation problems into simpler steps, making it an indispensable technique in calculus. Its versatility and ease of use make it a cornerstone for tackling a wide range of functions, ensuring accurate and efficient differentiation.
Result: dy/dx = -20x^{-6} + 2x^{-2}
By applying the power rule and the sum/difference rule, we have successfully differentiated the function y = 4x^{-5} - 2x^{-1}. The derivative, denoted as dy/dx, represents the rate of change of y with respect to x. Our calculations have shown that dy/dx = -20x^{-6} + 2x^{-2}. This result is crucial for understanding the behavior of the function, such as its increasing or decreasing intervals, local maxima and minima, and concavity. The derivative provides a powerful tool for analyzing and interpreting functions in calculus and various applications.
Interpreting the Derivative
Interpreting the derivative is essential for understanding the behavior of a function. The derivative, dy/dx = -20x^{-6} + 2x^{-2}, represents the instantaneous rate of change of the function y with respect to x. In simpler terms, it tells us how much y changes for a very small change in x. The sign of the derivative provides crucial information about the function's behavior: if dy/dx is positive, the function is increasing; if it's negative, the function is decreasing; and if it's zero, the function has a stationary point (a local maximum, minimum, or inflection point).
In our case, the derivative dy/dx = -20x^{-6} + 2x^{-2} can be further analyzed to understand the specific behavior of the function y = 4x^{-5} - 2x^{-1}. The term -20x^{-6} represents a negative rate of change that decreases rapidly as x increases, while the term 2x^{-2} represents a positive rate of change that decreases more slowly as x increases. By setting dy/dx = 0, we can find the critical points of the function, which are the points where the function's slope is zero. These points are potential locations of local maxima or minima. Furthermore, we can analyze the second derivative (the derivative of dy/dx) to determine the concavity of the function. A positive second derivative indicates that the function is concave up, while a negative second derivative indicates that it is concave down. Understanding these interpretations allows us to gain a comprehensive picture of the function's behavior, including its increasing and decreasing intervals, critical points, and concavity. This knowledge is invaluable in various applications, such as optimization problems, curve sketching, and modeling real-world phenomena.
Applications of Differentiation
Differentiation is a cornerstone of calculus with numerous practical applications across various fields. Understanding how to find and interpret derivatives is crucial for solving real-world problems in physics, engineering, economics, and more. In physics, differentiation is used to calculate velocity and acceleration from displacement functions. For example, if s(t) represents the position of an object at time t, the velocity v(t) is the first derivative of s(t), and the acceleration a(t) is the second derivative. This allows physicists to analyze the motion of objects and predict their behavior under different conditions.
In engineering, differentiation plays a critical role in optimization problems. Engineers often need to design systems or structures that maximize efficiency or minimize costs. Differentiation techniques, such as finding critical points and using the second derivative test, can help determine the optimal parameters for these designs. For instance, engineers might use differentiation to find the dimensions of a bridge that minimize stress or to optimize the performance of a circuit. In economics, differentiation is used to analyze marginal costs and revenues, helping businesses make decisions about pricing and production levels. The derivative of a cost function represents the marginal cost, which is the cost of producing one additional unit. Similarly, the derivative of a revenue function represents the marginal revenue, which is the revenue generated by selling one additional unit. By equating marginal cost and marginal revenue, businesses can find the production level that maximizes profit. Furthermore, differentiation is essential in mathematical modeling, where it helps in creating and analyzing differential equations that describe various phenomena. These equations are used to model population growth, chemical reactions, and many other processes. The versatility of differentiation makes it an indispensable tool in a wide array of applications, demonstrating its importance in both theoretical and practical contexts.
Conclusion
In conclusion, we have successfully differentiated the function y = 4x^{-5} - 2x^{-1} using the power rule and the sum/difference rule. The derivative, dy/dx = -20x^{-6} + 2x^{-2}, provides valuable information about the function's rate of change and behavior. Understanding differentiation is essential for calculus and its wide-ranging applications in various fields. This step-by-step guide ensures a clear understanding of the process and its significance.