Finding Dy/dx Integral Of Cos(t^3) With Variable Limits

Introduction

In this article, we will explore the process of finding the derivative of a function defined as an integral. Specifically, we will focus on the function y defined as the integral of cos(t^3) with respect to t, from the cube root of x to π/4. This problem involves applying the Fundamental Theorem of Calculus and the chain rule. Understanding these concepts is crucial for mastering calculus and its applications in various fields such as physics, engineering, and economics.

The Fundamental Theorem of Calculus provides a powerful connection between differentiation and integration. It essentially states that differentiation and integration are inverse processes. There are two parts to the theorem, but for this problem, we will primarily use the first part, which deals with the derivative of an integral with a variable limit of integration. Specifically, if we have a function F(x) defined as the integral of another function f(t) from a constant a to x, then the derivative of F(x) with respect to x is simply f(x). In mathematical notation, if F(x) = ∫[a to x] f(t) dt, then F'(x) = f(x). This theorem allows us to bypass the actual integration process when we are only interested in the derivative of the integral. However, in our case, the lower limit of integration is not a constant but a function of x, which introduces an additional layer of complexity requiring the chain rule.

The chain rule, on the other hand, is a fundamental concept in differential calculus that allows us to differentiate composite functions. A composite function is a function that is formed by applying one function to the result of another function. For example, if we have y = f(g(x)), then y is a composite function where g(x) is the inner function and f(x) is the outer function. The chain rule states that the derivative of y with respect to x is the product of the derivative of the outer function with respect to the inner function, and the derivative of the inner function with respect to x. Mathematically, this is expressed as dy/dx = f'(g(x)) * g'(x). The chain rule is essential when dealing with functions where the variable of differentiation is embedded within another function, such as in our integral problem where the lower limit of integration is a function of x.

To successfully tackle this problem, we will need to combine the Fundamental Theorem of Calculus with the chain rule. We will first rewrite the integral to have the variable limit of integration in the upper bound, which will require a sign change. Then, we will apply the Fundamental Theorem of Calculus to differentiate the integral, keeping in mind that the lower limit is a function of x. Finally, we will use the chain rule to account for the derivative of the lower limit function. This step-by-step approach will ensure that we correctly find the derivative dy/dx for the given function.

Problem Statement

Given the function:

$y = \int_{\sqrt[3]{x}}^{\pi / 4} \cos(t^3) dt$

Find $\frac{dy}{dx}$.

Solution

Step 1: Rewriting the Integral

To apply the Fundamental Theorem of Calculus more easily, we need to have the variable limit of integration as the upper bound. We can achieve this by reversing the limits of integration and changing the sign of the integral:

$y = -\int_{\pi / 4}^{\sqrt[3]{x}} \cos(t^3) dt$

This step is crucial because the Fundamental Theorem of Calculus in its standard form applies to integrals where the upper limit is the variable and the lower limit is a constant. By swapping the limits of integration and introducing a negative sign, we have prepared the integral for the direct application of the theorem.

Step 2: Applying the Fundamental Theorem of Calculus and the Chain Rule

Let $u = \sqrt[3]{x} = x^{\frac{1}{3}}$. Then, we can rewrite the integral as:

$y = -\int_{\pi / 4}^{u} \cos(t^3) dt$

Now, we can apply the Fundamental Theorem of Calculus along with the chain rule. The Fundamental Theorem of Calculus tells us that the derivative of an integral with respect to its upper limit is the integrand evaluated at that upper limit. The chain rule comes into play because the upper limit u is itself a function of x. Therefore, we need to multiply the derivative of the integral with respect to u by the derivative of u with respect to x.

Applying the Fundamental Theorem of Calculus, we get:

$\frac{dy}{du} = -\cos(u^3)$

This step directly utilizes the core concept of the Fundamental Theorem, which simplifies the differentiation of integrals. The theorem allows us to replace the integral with the integrand evaluated at the upper limit, making the differentiation process significantly easier.

Next, we need to find $\frac{du}{dx}$. Since $u = x^{\frac{1}{3}}$, we can differentiate u with respect to x using the power rule:

$\frac{du}{dx} = \frac{1}{3}x^{-\frac{2}{3}} = \frac{1}{3x^{\frac{2}{3}}}$

The power rule is a fundamental differentiation rule that states that the derivative of x raised to a power n is n times x raised to the power n-1. Applying this rule correctly is essential for finding the derivative of u with respect to x. The result, $\frac{1}{3}x^{-\frac{2}{3}}$, represents the rate of change of u with respect to x, which is a necessary component for applying the chain rule.

Now, we can apply the chain rule:

$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = -\cos(u^3) \cdot \frac{1}{3x^{\frac{2}{3}}}$

The chain rule allows us to connect the rate of change of y with respect to u and the rate of change of u with respect to x to find the overall rate of change of y with respect to x. This step combines the results from the previous steps, utilizing both the Fundamental Theorem of Calculus and the power rule, to express the derivative dy/dx in terms of u and x.

Step 3: Substituting Back

Finally, we substitute $u = \sqrt[3]{x}$ back into the expression:

$\frac{dy}{dx} = -\cos((\sqrt[3]{x})^3) \cdot \frac{1}{3x^{\frac{2}{3}}} = -\cos(x) \cdot \frac{1}{3x^{\frac{2}{3}}}$

This final substitution is crucial for expressing the derivative dy/dx solely in terms of x. By replacing u with its original expression, $\sqrt[3]{x}$, we obtain the final answer, which represents the derivative of the original function with respect to x. This step completes the solution and provides a clear and concise expression for the desired derivative.

Final Answer

Therefore,

$\frac{dy}{dx} = -\frac{\cos(x)}{3x^{\frac{2}{3}}}$

This result represents the derivative of the given integral function. It shows how the rate of change of y with respect to x depends on the cosine of x and a power of x. The negative sign indicates that as x increases, y decreases, at least in the regions where cos(x) is positive. The term $3x^{\frac{2}{3}}$ in the denominator suggests that the rate of change decreases as x increases, due to the increasing magnitude of the denominator. This final answer encapsulates the entire solution process, combining the application of the Fundamental Theorem of Calculus, the chain rule, and algebraic manipulation to find the derivative of the given function.

Conclusion

In this article, we successfully found the derivative $\frac{dy}{dx}$ for the function $y = \int_{\sqrt[3]{x}}^{\pi / 4} \cos(t^3) dt$ by applying the Fundamental Theorem of Calculus and the chain rule. The key steps involved rewriting the integral, differentiating using the Fundamental Theorem and chain rule, and substituting back to obtain the final result. This problem highlights the importance of understanding and applying these fundamental calculus concepts in solving complex problems. Mastering these techniques is crucial for further studies in mathematics and its applications in various scientific and engineering disciplines.

The process we followed demonstrates a systematic approach to solving differentiation problems involving integrals with variable limits. By carefully applying the Fundamental Theorem of Calculus and the chain rule, we were able to break down the problem into manageable steps and arrive at the correct solution. This approach can be applied to a wide range of similar problems, making it a valuable tool for any student of calculus. Furthermore, the problem illustrates the interconnectedness of different concepts in calculus, such as differentiation, integration, and the chain rule, emphasizing the importance of a holistic understanding of the subject. The final result, $\frac{dy}{dx} = -\frac{\cos(x)}{3x^{\frac{2}{3}}}$, provides a concise expression for the derivative, which can be further analyzed and used in various applications, such as finding critical points, analyzing the function's behavior, and solving related optimization problems.