Unveiling The Derivative Of Integral Function F(x) ∫₁ˣ³ T² Dt

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In the realm of calculus, the fundamental theorem of calculus serves as a cornerstone, bridging the concepts of differentiation and integration. This article delves into the application of this theorem to determine the derivative of a function defined as an integral. Specifically, we will explore the function f(x) = ∫₁ˣ³ t² dt and meticulously unravel the process of finding f'(x). Understanding this process not only solidifies the grasp of the fundamental theorem but also equips us with a powerful tool for tackling more intricate calculus problems.

Deciphering the Fundamental Theorem of Calculus

Before embarking on the solution, it's crucial to have a firm understanding of the Fundamental Theorem of Calculus. This theorem, in its essence, comprises two parts. The first part establishes the relationship between differentiation and integration, stating that if a function f is continuous on the interval [a, b], then the function F defined by F(x) = ∫ₐˣ f(t) dt is continuous on [a, b] and differentiable on (a, b), and its derivative is given by F'(x) = f(x). In simpler terms, differentiating the integral of a function returns the original function itself. However, the theorem's power truly shines when the upper limit of integration is not simply x but a function of x, as is the case in our problem.

The second part of the theorem provides a method for evaluating definite integrals. It states that if F is an antiderivative of f on the interval [a, b], then ∫ₐᵇ f(x) dx = F(b) - F(a). While this part is not directly used in solving for f'(x) in our problem, it is an essential component of the fundamental theorem and highlights the interconnectedness of differentiation and integration.

In the context of our problem, f(x) = ∫₁ˣ³ t² dt, the upper limit of integration is x³, a function of x. This necessitates the application of the chain rule in conjunction with the Fundamental Theorem of Calculus. The chain rule, a fundamental concept in differential calculus, allows us to differentiate composite functions. It states that if y = f(u) and u = g(x), then dy/dx = (dy/du) * (du/dx). This rule will be instrumental in handling the composite nature of our integral function.

Unveiling the Solution: Finding f'(x)

Now, let's embark on the journey of finding f'(x) for the function f(x) = ∫₁ˣ³ t² dt. This process involves a careful application of the Fundamental Theorem of Calculus and the chain rule. We'll break down the steps to ensure clarity and understanding.

  1. Identifying the Components: First, we need to recognize the different parts of our function. The integrand is t², and the upper limit of integration is x³. This upper limit, being a function of x, is the key to applying the chain rule.

  2. Applying the Fundamental Theorem: The Fundamental Theorem of Calculus tells us that if we were differentiating ∫₁ˣ t² dt, the result would simply be x². However, since the upper limit is x³, we need to account for this using the chain rule.

  3. Invoking the Chain Rule: Let's consider u = x³. Then, our function can be rewritten as f(x) = ∫₁ᵘ t² dt. Now, we can apply the chain rule: f'(x) = (d/du ∫₁ᵘ t² dt) * (du/dx).

  4. Differentiating the Integral: The first part, (d/du ∫₁ᵘ t² dt), is straightforward. According to the Fundamental Theorem of Calculus, this is simply u².

  5. Differentiating u: The second part, (du/dx), requires us to differentiate u = x³ with respect to x. This is a basic power rule differentiation, resulting in 3x².

  6. Putting it all Together: Now, we combine the results. f'(x) = u² * 3x². But we need to express the result in terms of x only. Recall that u = x³, so we substitute this back into the equation: f'(x) = (x³)² * 3x².

  7. Simplifying the Expression: Finally, we simplify the expression. (x³)² is x⁶, so we have f'(x) = x⁶ * 3x². Multiplying these terms together gives us the final answer: f'(x) = 3x⁸.

Therefore, the derivative of the function f(x) = ∫₁ˣ³ t² dt is f'(x) = 3x⁸. This result showcases the power of the Fundamental Theorem of Calculus in conjunction with the chain rule for differentiating functions defined as integrals with variable limits.

Elaborating on the Steps: A Deeper Dive

To further solidify your understanding, let's delve deeper into each step of the solution. We will explore the nuances and potential pitfalls to avoid.

Step 1: Identifying the Components

This step may seem trivial, but it's crucial for setting the stage for the rest of the solution. Correctly identifying the integrand (t²) and the upper limit of integration (x³) allows us to apply the appropriate rules and techniques. A common mistake is to overlook the fact that the upper limit is a function of x, which necessitates the chain rule.

Step 2: Applying the Fundamental Theorem

The Fundamental Theorem of Calculus is the cornerstone of this problem. It provides the direct link between differentiation and integration. However, it's essential to remember that the theorem applies directly when the upper limit of integration is simply x. When the upper limit is a function of x, as in our case, we must proceed to use the chain rule.

Step 3: Invoking the Chain Rule

The chain rule is the key to handling composite functions. In our context, the integral function is composed with the function x³. By introducing the variable u = x³, we effectively separate the composite function into simpler parts, making it easier to apply the differentiation rules. This step transforms the problem into a more manageable form.

Step 4: Differentiating the Integral

This step is a direct application of the Fundamental Theorem of Calculus. By considering u as the upper limit of integration, we can directly differentiate the integral ∫₁ᵘ t² dt with respect to u, obtaining u². This step highlights the elegance and power of the Fundamental Theorem.

Step 5: Differentiating u

This step involves a basic power rule differentiation. Differentiating u = x³ with respect to x yields 3x². This is a fundamental differentiation skill that is essential for many calculus problems. A solid grasp of basic differentiation rules is crucial for success in more complex problems.

Step 6: Putting it all Together

This step is where we combine the results from the previous steps. We multiply the derivative of the integral with respect to u (u²) by the derivative of u with respect to x (3x²). This application of the chain rule effectively accounts for the composite nature of the original function. It is important to maintain clarity in this step to avoid algebraic errors.

Step 7: Simplifying the Expression

The final step involves simplifying the expression to obtain the most concise and elegant form of the derivative. We substitute x³ back in for u and then use basic algebraic rules to simplify the expression. The final result, f'(x) = 3x⁸, is the derivative of the original function.

Potential Pitfalls and How to Avoid Them

While the solution process may seem straightforward, there are several potential pitfalls that students often encounter. Being aware of these pitfalls can help you avoid them and ensure accurate solutions.

  • Forgetting the Chain Rule: This is perhaps the most common mistake. Failing to recognize that the upper limit of integration is a function of x and not applying the chain rule will lead to an incorrect answer. Remember to always check the limits of integration carefully.
  • Misapplying the Fundamental Theorem: The Fundamental Theorem applies directly when the upper limit of integration is x. When it's a function of x, the chain rule must be used in conjunction with the theorem.
  • Algebraic Errors: Simplifying the final expression can be a source of errors. Pay close attention to the order of operations and the rules of exponents. Double-check your work to ensure accuracy.
  • Not Identifying the Integrand Correctly: Mistaking the integrand will obviously lead to an incorrect solution. Ensure you correctly identify the function being integrated.

By being mindful of these potential pitfalls and carefully following the steps outlined in this article, you can confidently tackle problems involving derivatives of integral functions.

The Significance and Applications

Understanding how to differentiate functions defined as integrals is not just an academic exercise. It has significant applications in various fields, including physics, engineering, and economics. For example, in physics, the derivative of a displacement function (defined as an integral of velocity) gives the acceleration. In engineering, these concepts are crucial for analyzing systems and predicting their behavior. In economics, they can be used to model marginal cost and revenue.

The ability to work with integral functions and their derivatives expands your mathematical toolkit and allows you to solve a wider range of problems in both theoretical and applied contexts. Mastering these concepts opens doors to more advanced topics in calculus and its applications.

Conclusion

In this article, we have meticulously explored the process of finding the derivative of the function f(x) = ∫₁ˣ³ t² dt. We have seen how the Fundamental Theorem of Calculus, in conjunction with the chain rule, provides a powerful framework for solving such problems. By understanding the underlying principles and practicing the steps, you can confidently tackle similar challenges. Remember to pay close attention to the details, avoid common pitfalls, and appreciate the significance of these concepts in various fields of study. The ability to differentiate functions defined as integrals is a valuable skill that will serve you well in your mathematical journey and beyond.