Derivative Of -7x³ - 4x⁸ A Step-by-Step Guide

In the realm of calculus, derivatives play a pivotal role. They are the cornerstone of understanding rates of change and are indispensable in diverse fields, from physics and engineering to economics and computer science. The derivative of a function, at its core, represents the instantaneous rate at which the function's output changes with respect to its input. This concept, while seemingly abstract, has profound practical implications. For instance, in physics, the derivative of an object's position with respect to time yields its velocity, and the derivative of velocity gives us acceleration. In economics, derivatives can be used to model marginal cost and revenue, providing businesses with invaluable insights into optimizing production and pricing strategies.

This article delves into the process of finding the derivative of the polynomial function -7x³ - 4x⁸. Polynomial functions, characterized by terms involving variables raised to non-negative integer powers, are fundamental in calculus and appear frequently in mathematical models. Mastering the differentiation of polynomial functions is therefore an essential skill for anyone venturing into calculus and its applications. We will meticulously break down the steps involved, leveraging the power rule and the constant multiple rule – two foundational concepts in differential calculus. The power rule provides a straightforward method for differentiating terms of the form xⁿ, while the constant multiple rule allows us to handle coefficients with ease. By applying these rules systematically, we will unravel the derivative of our given function, providing a clear and comprehensive explanation along the way.

Understanding the Power Rule

The power rule is a fundamental principle in calculus that simplifies the process of finding the derivative of power functions, which are expressions of the form xⁿ, where 'n' is any real number. This rule provides a direct and efficient method to calculate derivatives without resorting to the more cumbersome limit definition. The essence of the power rule lies in its elegant simplicity: to find the derivative of xⁿ, we multiply the term by the exponent 'n' and then reduce the exponent by one. Mathematically, this can be expressed as:

d/dx (xⁿ) = nxⁿ⁻¹

This deceptively simple formula is the key to unlocking the derivatives of a wide range of polynomial and algebraic functions. To truly grasp the power rule, let's delve into its mechanics with illustrative examples. Consider the function x². According to the power rule, its derivative is obtained by multiplying the term by the exponent 2, resulting in 2x, and then reducing the exponent by one, giving us 2x¹ or simply 2x. Similarly, the derivative of x³ can be found by multiplying by the exponent 3, yielding 3x, and reducing the exponent by one, resulting in 3x². These examples demonstrate the straightforward application of the power rule. However, the rule's versatility extends beyond simple integer exponents. It holds true for fractional and negative exponents as well. For instance, the derivative of √x, which can be written as x¹/², can be found using the power rule. Multiplying by the exponent 1/2 gives us (1/2)x, and reducing the exponent by one results in (1/2)x⁻¹/², which simplifies to 1/(2√x). This example showcases the power rule's ability to handle fractional exponents, expanding its applicability to a broader range of functions.

The power rule's significance lies in its ability to streamline differentiation. Instead of grappling with limits and complex algebraic manipulations, we can directly apply the rule to find derivatives. This not only saves time and effort but also reduces the likelihood of errors. The power rule is a cornerstone of differential calculus, and mastering its application is essential for anyone seeking to navigate the world of derivatives with confidence and proficiency.

Applying the Constant Multiple Rule

The constant multiple rule is another essential tool in the calculus toolkit, working in harmony with the power rule to simplify the differentiation process. This rule addresses how to handle constant coefficients that multiply a function. In essence, the constant multiple rule states that the derivative of a constant multiplied by a function is simply the constant multiplied by the derivative of the function. Mathematically, this can be expressed as:

d/dx [cf(x)] = c * d/dx [f(x)]

where 'c' is a constant and f(x) is a differentiable function. This rule provides a powerful mechanism for decoupling constants from the differentiation process, allowing us to focus on the function's derivative without being encumbered by the constant factor. To illustrate the constant multiple rule, let's consider the function 5x². Here, the constant is 5, and the function is x². According to the constant multiple rule, the derivative of 5x² is equal to 5 times the derivative of x². Applying the power rule to x², we find its derivative to be 2x. Therefore, the derivative of 5x² is 5 * 2x, which simplifies to 10x. This example demonstrates how the constant multiple rule allows us to treat the constant factor separately, making the differentiation process more manageable.

The constant multiple rule is particularly useful when dealing with polynomial functions, which often involve terms with constant coefficients. By applying this rule, we can differentiate each term individually, factoring out the constant before applying the power rule. This approach simplifies the overall differentiation process and reduces the chances of making algebraic errors. Moreover, the constant multiple rule extends beyond simple algebraic functions. It applies to a wide range of functions, including trigonometric, exponential, and logarithmic functions. This versatility makes it an indispensable tool in calculus, allowing us to differentiate complex expressions with greater ease and efficiency. The constant multiple rule, in conjunction with other differentiation rules, empowers us to tackle a vast array of calculus problems. Its ability to isolate constants from the differentiation process is a testament to its elegance and practicality, making it a cornerstone of differential calculus.

Differentiating -7x³

Now, let's apply the principles of the power rule and the constant multiple rule to differentiate the first term of our function, -7x³. This term comprises a constant coefficient, -7, and a power function, x³. Our strategy involves leveraging the constant multiple rule to separate the constant from the differentiation process and then applying the power rule to find the derivative of the power function.

According to the constant multiple rule, the derivative of -7x³ is equal to -7 times the derivative of x³. This allows us to focus solely on differentiating x³ without the distraction of the constant coefficient. Now, we turn our attention to the power rule, which provides a direct method for differentiating power functions. The power rule states that the derivative of xⁿ is nxⁿ⁻¹, where 'n' is the exponent. In our case, the exponent is 3. Applying the power rule to x³, we multiply the term by the exponent 3, resulting in 3x, and then reduce the exponent by one, giving us 3x². Therefore, the derivative of x³ is 3x².

Having found the derivative of x³, we now multiply it by the constant coefficient, -7, as dictated by the constant multiple rule. This gives us -7 * 3x², which simplifies to -21x². Thus, the derivative of -7x³ is -21x². This methodical approach, combining the constant multiple rule and the power rule, demonstrates the elegance and efficiency of these calculus principles. By breaking down the differentiation process into manageable steps, we can confidently tackle more complex functions. The ability to differentiate individual terms and then combine the results is a hallmark of calculus, allowing us to analyze and model a wide range of phenomena.

Differentiating -4x⁸

Next, let's tackle the second term of our function, -4x⁸. Similar to the previous step, we'll employ the constant multiple rule and the power rule to find its derivative. This term also consists of a constant coefficient, -4, and a power function, x⁸. The constant multiple rule will again allow us to isolate the constant and focus on differentiating the power function.

Applying the constant multiple rule to -4x⁸, we recognize that its derivative is equal to -4 times the derivative of x⁸. This simplifies our task, allowing us to concentrate on finding the derivative of x⁸. Now, we invoke the power rule, which states that the derivative of xⁿ is nxⁿ⁻¹. In this instance, the exponent is 8. Applying the power rule to x⁸, we multiply the term by the exponent 8, resulting in 8x, and then reduce the exponent by one, giving us 8x⁷. Therefore, the derivative of x⁸ is 8x⁷.

Having determined the derivative of x⁸, we now multiply it by the constant coefficient, -4, as per the constant multiple rule. This yields -4 * 8x⁷, which simplifies to -32x⁷. Consequently, the derivative of -4x⁸ is -32x⁷. This step further reinforces the effectiveness of combining the constant multiple rule and the power rule. By systematically applying these principles, we can confidently differentiate terms involving constant coefficients and power functions. This methodical approach is crucial for tackling more intricate calculus problems, where functions may consist of multiple terms and require careful application of differentiation rules. The ability to break down complex expressions into simpler components and then differentiate them individually is a cornerstone of calculus proficiency.

Combining the Derivatives

Having found the derivatives of both terms in our original function, -7x³ and -4x⁸, we now combine them to obtain the derivative of the entire function. Recall that the derivative of a sum (or difference) of functions is simply the sum (or difference) of their individual derivatives. This principle allows us to differentiate complex functions term by term, making the process more manageable.

We previously determined that the derivative of -7x³ is -21x² and the derivative of -4x⁸ is -32x⁷. Therefore, to find the derivative of the entire function, -7x³ - 4x⁸, we simply add these two derivatives together:

d/dx (-7x³ - 4x⁸) = d/dx (-7x³) + d/dx (-4x⁸)

Substituting the derivatives we calculated earlier, we get:

-21x² + (-32x⁷)

This can be simplified to:

-21x² - 32x⁷

Thus, the derivative of -7x³ - 4x⁸ is -21x² - 32x⁷. This result represents the instantaneous rate of change of the function at any given value of x. The negative signs indicate that the function is decreasing as x increases in certain intervals, while the exponents indicate the rate at which this change occurs. This final step demonstrates the power of calculus in providing insights into the behavior of functions. By combining the derivatives of individual terms, we can understand the overall rate of change of a function and its relationship to the input variable. This knowledge is invaluable in various applications, from optimizing engineering designs to modeling economic trends.

Final Answer

In conclusion, by systematically applying the constant multiple rule and the power rule, we have successfully found the derivative of the function -7x³ - 4x⁸. The derivative, as we have shown, is:

-21x² - 32x⁷

This result encapsulates the instantaneous rate of change of the original function at any given point. The process we have undertaken highlights the fundamental principles of differential calculus and their power in analyzing the behavior of functions. Mastering these principles is crucial for anyone seeking to delve deeper into mathematics, physics, engineering, or any field that relies on mathematical modeling. The derivative, as a concept, is not merely an abstract mathematical construct; it is a tool that unlocks a deeper understanding of the world around us. From calculating velocities and accelerations to optimizing industrial processes, derivatives play a vital role in countless applications. The ability to find derivatives efficiently and accurately is therefore an invaluable skill.

This comprehensive guide has provided a step-by-step approach to differentiating a polynomial function, emphasizing the importance of the power rule and the constant multiple rule. By understanding and applying these rules, you can confidently tackle a wide range of differentiation problems. Remember, practice is key to mastering calculus. Work through numerous examples, and you will find yourself increasingly comfortable with the concepts and techniques involved. The journey into calculus is a rewarding one, opening up new avenues of mathematical understanding and problem-solving prowess.