Finding The Derivative Of F(x) = 1/(9-x^2)^6 A Calculus Tutorial

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This article delves into the process of finding the derivative of a complex function, specifically f(x) = 1/(9-x2)6. We'll break down the steps involved, utilizing key calculus principles such as the chain rule and power rule, to arrive at the solution for f'(x). This exploration is crucial for anyone seeking to enhance their understanding of differential calculus and its applications in various fields. Understanding how to differentiate complex functions like this is a fundamental skill in calculus, opening doors to solving a wide range of problems in physics, engineering, economics, and more. By mastering these techniques, you'll gain a deeper appreciation for the power and versatility of calculus.

Understanding the Function and the Challenge

Before we dive into the differentiation process, let's first understand the structure of the function f(x) = 1/(9-x2)6. This function is a composite function, meaning it's a function within a function. We have a rational function raised to a power. The presence of this composition necessitates the use of the chain rule, a vital tool in differential calculus. The chain rule allows us to differentiate composite functions by breaking them down into simpler parts and applying differentiation rules sequentially. It essentially states that the derivative of a composite function is the product of the derivatives of its outer and inner functions. In our case, the outer function is something to the power of -6, and the inner function is (9-x^2). Recognizing this structure is the first step in successfully finding the derivative. The challenge lies in carefully applying the chain rule and power rule in conjunction, ensuring we account for all parts of the function. This requires a systematic approach and a solid understanding of the underlying principles of differentiation.

Applying the Chain Rule and Power Rule

The heart of finding f'(x) lies in the application of the chain rule and the power rule. The chain rule, as mentioned earlier, is our key to tackling composite functions. The power rule, on the other hand, helps us differentiate terms raised to a power. It states that the derivative of x^n is nx^(n-1). In our case, we can rewrite f(x) as (9-x2)-6. This form makes it clearer how to apply the power rule. We bring the exponent (-6) down and multiply it with the function, then reduce the exponent by 1. This gives us -6*(9-x2)-7. However, we're not done yet! This is where the chain rule kicks in. We need to multiply this result by the derivative of the inner function, which is (9-x^2). The derivative of (9-x^2) is -2x. Therefore, we multiply -6*(9-x2)-7 by -2x. This entire process demonstrates the elegance and power of these calculus rules in action. The combination of the chain rule and power rule allows us to systematically differentiate even complex functions.

Step-by-Step Differentiation Process

Let's break down the differentiation process into clear, manageable steps:

  1. Rewrite the function: Begin by rewriting f(x) = 1/(9-x2)6 as f(x) = (9-x2)-6. This form makes the application of the power rule more straightforward.
  2. Apply the power rule and chain rule: Using the power rule, we bring down the exponent (-6) and multiply it with the function, reducing the exponent by 1: -6*(9-x2)-7.
  3. Differentiate the inner function: Now, we need to find the derivative of the inner function, (9-x^2). The derivative of 9 (a constant) is 0, and the derivative of -x^2 is -2x. Therefore, the derivative of the inner function is -2x.
  4. Multiply the results: Finally, we multiply the result from step 2 with the derivative of the inner function from step 3: -6*(9-x2)-7 * (-2x).
  5. Simplify the expression: Simplify the expression to obtain the final derivative. This involves multiplying the constants and rewriting the negative exponent as a fraction.

Following these steps meticulously ensures accuracy and clarity in the differentiation process. Each step builds upon the previous one, leading us to the final solution.

Simplifying the Result and Finding f'(x)

After applying the chain rule and power rule, we arrive at the expression: f'(x) = -6(9-x2)-7 * (-2x)*. Now, it's time to simplify this expression to its most elegant form. First, we multiply the constants: -6 * -2x = 12x. This gives us f'(x) = 12x * (9-x2)-7. To further simplify, we can rewrite the term with the negative exponent as a fraction. Recall that a^(-n) = 1/a^n. Therefore, (9-x2)-7 can be rewritten as 1/(9-x2)7. Substituting this back into our expression, we get f'(x) = 12x / (9-x2)7. This is the simplified form of the derivative. It represents the instantaneous rate of change of the function f(x) at any given point x. The process of simplification is crucial in mathematics as it allows us to present the answer in a clear and concise manner, making it easier to interpret and use in further calculations.

The Final Answer and Its Significance

After meticulous application of the chain rule, power rule, and simplification techniques, we have arrived at the final answer: f'(x) = 12x / (9-x2)7. This is the derivative of the function f(x) = 1/(9-x2)6. This result represents the slope of the tangent line to the curve of f(x) at any point x. In other words, it tells us how the function f(x) is changing at a particular value of x. The derivative is a fundamental concept in calculus with wide-ranging applications. It's used to find maximum and minimum values of functions, analyze the behavior of curves, solve optimization problems, and much more. Understanding the derivative is essential for anyone working in fields that involve mathematical modeling, such as physics, engineering, economics, and computer science. The ability to find and interpret derivatives is a powerful tool for understanding and manipulating mathematical relationships.

Conclusion: Mastering Differentiation Techniques

In this article, we've successfully navigated the process of finding the derivative of the function f(x) = 1/(9-x2)6. We've demonstrated the importance of the chain rule and power rule in differentiating composite functions. We've also emphasized the need for careful simplification to arrive at the final answer: f'(x) = 12x / (9-x2)7. This journey highlights the core principles of differential calculus and their application in solving complex problems. Mastering these techniques is crucial for anyone seeking a deeper understanding of mathematics and its applications in various scientific and engineering disciplines. The ability to differentiate functions is a fundamental skill that opens doors to a wide range of problem-solving capabilities. By practicing and applying these techniques, you'll not only improve your mathematical proficiency but also gain a valuable tool for analyzing and understanding the world around you.