Finding Derivatives Of Integrals A Step By Step Guide

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Understanding Derivatives and the Fundamental Theorem of Calculus

The core concept we'll be exploring in this article is the derivative, a fundamental tool in calculus that measures the instantaneous rate of change of a function. In simpler terms, the derivative tells us how much a function's output changes for a tiny change in its input. To tackle the problem at hand, we'll also need to understand the Fundamental Theorem of Calculus, which establishes a crucial link between differentiation and integration. This theorem has two parts, but the one we're most interested in here is the first part, which states that if we have a function defined as an integral, its derivative can be found relatively easily.

Delving Deeper into the First Fundamental Theorem of Calculus

Let's break down the First Fundamental Theorem of Calculus a bit further. It essentially says that if we have a function F(x) defined as the integral of another function f(t) from a constant a to a variable x, that is:

F(x) = ∫ₐˣ f(t) dt

Then, the derivative of F(x) with respect to x is simply f(x):

F'(x) = d/dx [∫ₐˣ f(t) dt] = f(x)

This theorem provides a powerful shortcut for finding derivatives of functions defined as integrals. Instead of having to evaluate the integral and then differentiate the result, we can directly substitute the upper limit of integration into the integrand. However, we must be mindful of the fact that this theorem applies directly when the upper limit of integration is simply x. When the upper limit is a function of x, like x² in our problem, we need to use the chain rule in conjunction with the theorem. This is a crucial point to remember as we move forward.

Chain Rule and Its Importance

Before we jump into solving the problem, let's briefly discuss the chain rule. The chain rule is another essential concept in calculus that helps us find the derivative of composite functions – functions that are made up of other functions. If we have a function y = f(g(x)), where g(x) is a function of x and f(x) is a function of g(x), then the chain rule states that the derivative of y with respect to x is:

dy/dx = f'(g(x)) * g'(x)

In simpler terms, we take the derivative of the outer function f with respect to the inner function g(x), and then multiply it by the derivative of the inner function g(x) with respect to x. The chain rule is indispensable when dealing with composite functions, and it will play a vital role in solving our derivative problem since the upper limit of our integral is x², a function of x. Understanding and correctly applying the chain rule is key to accurately finding the derivative in this scenario.

Applying the Concepts to the Problem

Now, let's apply these concepts to the problem at hand. We are given the function:

A(x) = ∫₁ˣ² 2t dt

We are asked to find A'(x), the derivative of A(x) with respect to x. Notice that the upper limit of integration is x², a function of x, not just x. This is where the chain rule comes into play. We can think of A(x) as a composite function, where the outer function is the integral and the inner function is x². To find the derivative, we'll use the First Fundamental Theorem of Calculus in conjunction with the chain rule.

Step-by-Step Solution

Let's break down the solution step-by-step:

  1. Apply the First Fundamental Theorem of Calculus and the Chain Rule:

    According to the First Fundamental Theorem of Calculus, if the upper limit of integration was simply x, the derivative would be 2x. However, since the upper limit is x², we need to use the chain rule. We first substitute x² into the integrand (2t), replacing t with x², which gives us 2x². Then, we multiply this by the derivative of the inner function, x², which is 2x. Therefore:

    A'(x) = 2(x²) * d/dx (x²)

  2. Calculate the derivative of x²:

    The derivative of x² with respect to x is 2x. This is a basic power rule application in differentiation.

  3. Substitute and Simplify:

    Now, substitute the derivative of x² back into the equation:

    A'(x) = 2(x²) * (2x)

    Multiply the terms to simplify the expression:

    A'(x) = 4x³

Therefore, the derivative of A(x) is 4x³.

Final Answer and the Importance of Showing Work

Thus, we have found that:

A'(x) = 4x³

This matches the format requested in the problem, where we needed to find the coefficient and the exponent of x. In this case, the coefficient is 4 and the exponent is 3. It's worth emphasizing the importance of showing each step in the solution process. Not only does it help in understanding the logic and reasoning behind the answer, but it also makes it easier to identify any potential errors. In calculus, a small mistake in one step can propagate through the entire solution, leading to an incorrect final answer. By meticulously showing our work, we can minimize the chances of making such errors and ensure the accuracy of our results.

Conclusion

In this article, we've explored the process of finding the derivative of a function defined as an integral using the Fundamental Theorem of Calculus and the chain rule. We've seen how these two powerful tools can be combined to solve complex problems in calculus. Understanding these concepts is crucial for anyone studying calculus, as they form the foundation for many other advanced topics. By breaking down the problem into smaller, manageable steps and carefully applying the relevant theorems and rules, we can successfully find the derivative and simplify our answer. Remember, practice is key to mastering these concepts. The more problems you solve, the more comfortable you'll become with applying the Fundamental Theorem of Calculus and the chain rule. So, keep practicing, keep exploring, and keep learning!