Derivative Of Tan(2x+3) A Step-by-Step Calculus Guide
In this guide, we will explore the derivative of tan(2x+3), a common problem in calculus. This step-by-step explanation aims to provide a clear and thorough understanding of how to approach such derivatives, making it easier for students and calculus enthusiasts to grasp the underlying concepts and techniques involved. We will delve into the fundamental rules of differentiation, specifically the chain rule, which is crucial for solving this type of problem. By breaking down each step, we ensure that you not only understand the solution but also the reasoning behind it. Whether you're a student tackling calculus homework or simply someone looking to refresh your knowledge, this guide will equip you with the skills to confidently handle derivatives of trigonometric functions with composite arguments.
Understanding the Basics
Before we dive into the specifics of finding the derivative of tan(2x+3), it’s essential to have a solid grasp of the fundamental concepts. This includes understanding what a derivative represents and the basic rules of differentiation. The derivative of a function measures the instantaneous rate of change of the function with respect to its variable. In simpler terms, it tells us how much the function's output changes for a tiny change in its input. For example, if we're dealing with a function that represents the position of an object over time, its derivative would represent the object's velocity at any given moment. This concept is foundational in calculus and has wide-ranging applications in physics, engineering, economics, and more.
Differentiation, the process of finding the derivative, relies on a set of rules that allow us to calculate derivatives of various types of functions. Some of the most basic rules include the power rule, which states that the derivative of x^n is nx^(n-1), and the constant multiple rule, which says that the derivative of cf(x) is cf'(x), where c is a constant. Additionally, we have the sum and difference rules, which tell us how to differentiate sums and differences of functions. However, when dealing with trigonometric functions or composite functions, we need to employ more specialized rules such as the derivatives of trigonometric functions and the chain rule. Mastering these basics is the cornerstone for tackling more complex differentiation problems like the one we're addressing in this guide.
The Chain Rule
The chain rule is a fundamental concept in calculus that allows us to find the derivatives of composite functions. A composite function is essentially a function within a function, like our example of tan(2x+3). The chain rule states that if we have a composite function y = f(g(x)), then its derivative, dy/dx, is given by dy/du * du/dx, where u = g(x). In simpler terms, we take the derivative of the outer function with respect to the inner function, and then multiply it by the derivative of the inner function with respect to x. This rule is crucial for differentiating functions where one function is nested inside another, which is a common occurrence in calculus.
To illustrate the chain rule, consider a simpler example, such as sin(x^2). Here, the outer function is sin(u) and the inner function is u = x^2. To find the derivative, we first differentiate the outer function with respect to u, which gives us cos(u). Then, we differentiate the inner function with respect to x, which gives us 2x. Finally, we multiply these two results together, giving us cos(u) * 2x. Substituting x^2 back in for u, we get the final derivative: 2x * cos(x^2). Understanding this process is key to applying the chain rule effectively in more complex scenarios. In the case of tan(2x+3), we'll see how the chain rule allows us to systematically break down the problem and find the derivative step by step.
Derivative of tan(x)
Before we can tackle the derivative of tan(2x+3), it's crucial to know the derivative of the basic tangent function, tan(x). The derivative of tan(x) is a fundamental result in calculus and is given by sec^2(x). This result is derived using the quotient rule and the derivatives of sine and cosine. Recall that tan(x) is defined as sin(x)/cos(x). To find its derivative, we apply the quotient rule, which states that the derivative of (u/v) is (v(du/dx) - u(dv/dx)) / v^2.
Applying the quotient rule to tan(x) = sin(x)/cos(x), we let u = sin(x) and v = cos(x). The derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x). Substituting these into the quotient rule formula, we get: (cos(x) * cos(x) - sin(x) * (-sin(x))) / (cos^2(x)). Simplifying this expression, we have (cos^2(x) + sin^2(x)) / (cos^2(x)). From the Pythagorean trigonometric identity, we know that cos^2(x) + sin^2(x) = 1. Therefore, the expression simplifies to 1 / cos^2(x). By definition, 1 / cos^2(x) is equal to sec^2(x). Hence, the derivative of tan(x) is sec^2(x). This foundational knowledge is essential because, in the chain rule application, we'll be using this result as the derivative of the outer function.
Step-by-Step Solution for tan(2x+3)
Now, let's dive into the step-by-step solution for finding the derivative of tan(2x+3). This is where we'll apply the chain rule, building on our understanding of the derivative of tan(x). The function we're dealing with is a composite function, where the outer function is tan(u) and the inner function is u = 2x+3. Our goal is to find the derivative of this composite function with respect to x, which we denote as d/dx [tan(2x+3)].
Step 1: Identify the Outer and Inner Functions The first step in applying the chain rule is to clearly identify the outer and inner functions. As mentioned, the outer function is the tangent function, tan(u), and the inner function is the expression inside the tangent, which is u = 2x+3. Recognizing these components is crucial because it dictates how we apply the chain rule formula. The outer function is what we directly apply the basic derivative rules to, while the inner function will require its own derivative to be calculated.
Step 2: Differentiate the Outer Function Next, we differentiate the outer function, tan(u), with respect to u. We know from the previous discussion that the derivative of tan(x) is sec^2(x). Therefore, the derivative of tan(u) with respect to u is sec^2(u). This step is a direct application of a known derivative rule. It's important to remember that we are differentiating with respect to u at this stage, not x, which is why we keep u in the expression. This result, sec^2(u), will be one part of our final answer, but we still need to account for the inner function.
Step 3: Differentiate the Inner Function Now, we need to differentiate the inner function, u = 2x+3, with respect to x. This is a straightforward differentiation using the power rule and the constant rule. The derivative of 2x with respect to x is 2, and the derivative of the constant 3 is 0. Therefore, the derivative of 2x+3 with respect to x is simply 2. This result represents the rate of change of the inner function with respect to x and is the second crucial component we need for the chain rule.
Step 4: Apply the Chain Rule Now that we have the derivative of the outer function with respect to u, which is sec^2(u), and the derivative of the inner function with respect to x, which is 2, we can apply the chain rule. The chain rule states that d/dx [tan(2x+3)] = d/du [tan(u)] * du/dx. Substituting the results we found in steps 2 and 3, we get:d/dx [tan(2x+3)] = sec^2(u) * 2. This is almost our final answer, but we need to express it in terms of x.
Step 5: Substitute Back The final step is to substitute the inner function, u = 2x+3, back into our expression. This replaces u with its equivalent expression in terms of x, giving us the derivative of the original function with respect to x. Substituting 2x+3 for u in sec^2(u) * 2, we get:d/dx [tan(2x+3)] = 2 * sec^2(2x+3). This is the final derivative of tan(2x+3). It represents the instantaneous rate of change of the function tan(2x+3) with respect to x.
Conclusion
In conclusion, finding the derivative of tan(2x+3) involves a methodical application of the chain rule, combined with the knowledge of basic derivative rules. By breaking down the problem into smaller, manageable steps, we can clearly see how each component contributes to the final solution. The process involves identifying the outer and inner functions, differentiating each separately, and then combining the results using the chain rule. This approach is not only effective for this specific problem but also serves as a template for tackling other composite functions in calculus.
We started by understanding the basics of derivatives and the significance of the chain rule. We then recalled the derivative of the tangent function, which is essential for solving this problem. The step-by-step solution involved identifying the outer and inner functions, differentiating them individually, applying the chain rule, and finally substituting back to get the derivative in terms of x. This methodical approach ensures that we don't miss any steps and arrive at the correct answer.
Mastering the chain rule is crucial for success in calculus, as it is a fundamental tool for differentiating composite functions. By practicing more problems and understanding the underlying concepts, you can confidently handle a wide range of derivatives. This guide provides a solid foundation for understanding and applying the chain rule, making calculus more approachable and less daunting. Whether you are a student or someone looking to refresh your calculus skills, the principles and techniques discussed here will be invaluable in your mathematical journey.
