Derivative Of Sin((πx)^2) A Step-by-Step Guide

In the realm of calculus, derivatives play a crucial role in understanding the rate at which a function changes. Among various functions, trigonometric functions hold significant importance due to their periodic nature and widespread applications in physics, engineering, and other scientific disciplines. In this comprehensive guide, we delve into the process of finding the derivative of a specific trigonometric function, ${y = \sin((\pi x)^2)}$. This exploration will not only enhance your understanding of derivative techniques but also provide you with a step-by-step approach to tackling similar problems. Mastering the derivatives of trigonometric functions is an essential skill for anyone delving into calculus. This guide focuses specifically on finding the derivative of ${y = \sin((\pi x)^2)}$, a function that combines trigonometric and polynomial elements, making it an excellent example for illustrating the chain rule and other derivative techniques. By the end of this article, you'll have a solid understanding of how to approach such problems and a step-by-step method to solve them.

Understanding the Function: ${y = \sin((\pi x)^2)}$

Before we dive into the process of differentiation, let's take a moment to understand the function we're dealing with. The function ${y = \sin((\pi x)^2)}$ is a composite function, meaning it's a function within a function. Here, we have the sine function, which is a trigonometric function, and inside the sine function, we have ${(\pi x)^2}$, which is a polynomial function. To find the derivative of such a composite function, we need to apply the chain rule, a fundamental concept in calculus.

To fully grasp the intricacies of this function, let's break it down into its core components. At its heart, we have the sine function, a cornerstone of trigonometry. The sine function, denoted as ${\sin(x)}$, oscillates between -1 and 1, creating a wave-like pattern. Its derivative, ${\cos(x)}$, is equally important and forms the basis for many calculus operations involving trigonometric functions. In our case, the sine function is not acting on a simple variable ${x}$, but rather on a more complex expression, ${(\pi x)^2}$. This expression represents a polynomial function, specifically a quadratic function scaled by the constant ${\pi^2}$. The presence of this inner function makes our task a bit more challenging, requiring the application of the chain rule. The chain rule allows us to differentiate composite functions by systematically addressing each layer of the function. It essentially states that the derivative of a composite function is the product of the derivative of the outer function evaluated at the inner function and the derivative of the inner function itself. This may sound complex, but we'll break it down step by step in the following sections.

Applying the Chain Rule

The chain rule is the key to finding the derivative of ${y = \sin((\pi x)^2)}$. It states that if we have a composite function ${y = f(g(x))}$, then its derivative ${y'}$ is given by:

${y' = f'(g(x)) \cdot g'(x)}$

In our case, we can identify the outer function as ${f(u) = \sin(u)}$ and the inner function as ${g(x) = (\pi x)^2}$. Therefore, we need to find the derivatives of both these functions.

Let's start by identifying the outer and inner functions in our specific case. The outer function, which we'll call ${f(u)}$, is the sine function: ${f(u) = \sin(u)}$. This is the function that acts on the entire expression inside the parentheses. The inner function, which we'll call ${g(x)}$, is the expression inside the sine function: ${g(x) = (\pi x)^2}$. This function is a polynomial function, and it's the argument of the sine function. Now that we've identified the outer and inner functions, we can move on to finding their derivatives. The derivative of the outer function, ${f'(u)}$, is the derivative of ${\sin(u)}$, which is ${\cos(u)}$. Remember, we're taking the derivative with respect to ${u}$, so we treat ${u}$ as the variable. The derivative of the inner function, ${g'(x)}$, requires a bit more work. We have ${g(x) = (\pi x)^2}$, which can be rewritten as ${g(x) = \pi^2 x^2}$. To find its derivative, we apply the power rule, which states that the derivative of ${x^n}$ is ${nx^{n-1}}$. Applying the power rule and the constant multiple rule (which states that the derivative of a constant times a function is the constant times the derivative of the function), we get ${g'(x) = 2\pi^2 x}$. Now that we have the derivatives of both the outer and inner functions, we can apply the chain rule. The chain rule tells us to multiply the derivative of the outer function (evaluated at the inner function) by the derivative of the inner function. In other words, we need to multiply ${f'(g(x))}$ by ${g'(x)}$.

Differentiating the Outer Function: ${f(u) = \sin(u)}$

The derivative of the outer function, ${f(u) = \sin(u)}$, is a fundamental derivative in calculus. We know that:

${f'(u) = \frac{d}{du} \sin(u) = \cos(u)}$

This is a standard result that you'll encounter frequently when dealing with trigonometric functions. Now, we need to evaluate this derivative at the inner function, ${g(x) = (\pi x)^2}$, which means replacing ${u}$ with ${(\pi x)^2}$. This gives us:

${f'(g(x)) = \cos((\pi x)^2)}$

The derivative of the sine function is the cosine function, a fundamental relationship in calculus. Understanding this relationship is crucial for differentiating more complex trigonometric functions. The derivative of ${\sin(u)}$ with respect to ${u}$ is simply ${\cos(u)}$. This is a well-established rule that you'll encounter repeatedly in calculus problems. However, in our case, we're not dealing with a simple ${u}$, but rather with the inner function ${g(x) = (\pi x)^2}$. This means we need to substitute ${(\pi x)^2}$ for ${u}$ in the derivative of the outer function. This substitution gives us ${\cos((\pi x)^2)}$, which is the derivative of the outer function evaluated at the inner function. This step is a critical application of the chain rule, ensuring that we account for the effect of the inner function on the overall derivative. Remember, the chain rule requires us to work from the outside in, first differentiating the outer function and then addressing the inner function. By correctly substituting the inner function into the derivative of the outer function, we're setting ourselves up for the next step, which involves differentiating the inner function itself. This systematic approach is key to mastering the chain rule and successfully differentiating composite functions.

Differentiating the Inner Function: ${g(x) = (\pi x)^2}$

Next, we need to find the derivative of the inner function, ${g(x) = (\pi x)^2}$. We can rewrite this as:

${g(x) = \pi^2 x^2}$

Now, applying the power rule, we get:

${g'(x) = \frac{d}{dx} (\pi^2 x^2) = 2\pi^2 x}$

The inner function, ${g(x) = (\pi x)^2}$, is a polynomial function, and differentiating it requires the application of the power rule and the constant multiple rule. Before we apply these rules, it's helpful to rewrite the function as ${g(x) = \pi^2 x^2}$. This makes it clearer that we have a constant, ${\pi^2}$, multiplied by a power of ${x}$. The power rule states that the derivative of ${x^n}$ is ${nx^{n-1}}$. In our case, ${n = 2}$, so the derivative of ${x^2}$ is ${2x}$. The constant multiple rule states that the derivative of a constant times a function is the constant times the derivative of the function. Applying this rule, we multiply the constant ${\pi^2}$ by the derivative of ${x^2}$, which is ${2x}$. This gives us ${2\pi^2 x}$, which is the derivative of the inner function. This derivative represents the rate of change of the inner function with respect to ${x}$. It's a crucial component in the chain rule, as it accounts for how the inner function's change affects the overall function's change. By correctly differentiating the inner function, we've completed the second key step in applying the chain rule. We now have both the derivative of the outer function (evaluated at the inner function) and the derivative of the inner function itself. The next step is to combine these two derivatives using the chain rule to find the derivative of the entire composite function.

Combining the Derivatives

Now that we have ${f'(g(x)) = \cos((\pi x)^2)}$ and ${g'(x) = 2\pi^2 x}$, we can apply the chain rule:

${y' = f'(g(x)) \cdot g'(x) = \cos((\pi x)^2) \cdot 2\pi^2 x}$

Therefore, the derivative of ${y = \sin((\pi x)^2)}$ is:

${y'(x) = 2\pi^2 x \cos((\pi x)^2)}$

We've successfully found the derivative of our composite function by systematically applying the chain rule. The final step is to combine the derivatives we calculated in the previous sections. We have ${f'(g(x)) = \cos((\pi x)^2)}$, which is the derivative of the outer function evaluated at the inner function, and ${g'(x) = 2\pi^2 x}$, which is the derivative of the inner function. The chain rule tells us to multiply these two derivatives together to get the derivative of the entire composite function. This gives us ${y' = \cos((\pi x)^2) \cdot 2\pi^2 x}$. We can rewrite this as ${y'(x) = 2\pi^2 x \cos((\pi x)^2)}$, which is the final derivative of ${y = \sin((\pi x)^2)}$. This result tells us how the function ${y}$ changes with respect to changes in ${x}$. It's a valuable piece of information that can be used in various applications, such as finding the slope of the tangent line to the function at a given point, determining the function's critical points, and analyzing its concavity. By mastering the chain rule and applying it systematically, we've successfully navigated the complexities of differentiating a composite trigonometric function.

Conclusion

Finding the derivative of ${y = \sin((\pi x)^2)}$ involves a careful application of the chain rule. By breaking down the function into its outer and inner components, differentiating each separately, and then combining the results, we arrive at the derivative:

${y'(x) = 2\pi^2 x \cos((\pi x)^2)}$

This exercise demonstrates the power and elegance of calculus in handling complex functions. The process of finding the derivative of ${y = \sin((\pi x)^2)}$ highlights the importance of the chain rule in calculus. This rule allows us to differentiate composite functions, which are functions within functions, by systematically addressing each layer. We started by identifying the outer and inner functions, then differentiated each separately, and finally combined the results using the chain rule formula. This step-by-step approach is crucial for avoiding errors and ensuring accuracy. The derivative we found, ${y'(x) = 2\pi^2 x \cos((\pi x)^2)}$, provides valuable information about the original function. It tells us the rate at which the function is changing at any given point. This information can be used to analyze the function's behavior, such as finding its maximum and minimum values, determining its intervals of increase and decrease, and sketching its graph. Understanding derivatives is fundamental to many areas of mathematics, science, and engineering. It allows us to model and analyze dynamic systems, optimize processes, and solve a wide range of problems. By mastering the techniques presented in this guide, you'll be well-equipped to tackle more complex calculus problems and apply these concepts in various fields. Remember, practice is key to solidifying your understanding. Work through similar examples and challenge yourself with more complex functions to further develop your skills in differentiation. The chain rule is a powerful tool, and with consistent practice, you'll become proficient in its application.