Finding The Derivative Of Y=(3x^-3 + 2x)^3 A Step-by-Step Guide
In the realm of calculus, derivatives serve as a cornerstone for understanding rates of change and the behavior of functions. This article delves into the process of finding the derivative of the function y = (3x^-3 + 2x)^3 with respect to x. We will break down the problem step-by-step, employing fundamental rules of differentiation, including the power rule and the chain rule. This exploration not only provides a solution but also aims to foster a deeper understanding of the underlying principles.
Understanding the Building Blocks: Power Rule and Chain Rule
Before we embark on the journey of differentiating the given function, let's solidify our understanding of the essential tools: the power rule and the chain rule. These rules are the cornerstones of differentiation and will be instrumental in solving our problem.
The Power Rule
The power rule is a fundamental principle that governs the differentiation of power functions. A power function is simply a variable raised to a constant exponent, represented as x^n, where n is any real number. The power rule elegantly states that the derivative of x^n with respect to x is n times x raised to the power of n-1. Mathematically, this can be expressed as:
d/dx (x^n) = nx^(n-1)
This rule is widely applicable and forms the basis for differentiating various polynomial and algebraic expressions. It allows us to systematically reduce the exponent of the variable while multiplying by the original exponent. For example, the derivative of x^3 is 3x^2, and the derivative of x^-2 is -2x^-3.
The power rule is not just a formula to be memorized; it has a solid foundation in the definition of the derivative and the concept of limits. It provides a shortcut for finding the derivative of power functions, saving us the need to resort to the limit definition every time. Understanding the power rule is crucial for mastering differentiation and tackling more complex problems.
The Chain Rule
The chain rule is a powerful tool for differentiating composite functions. A composite function is a function that is formed by applying one function to the result of another function. In simpler terms, it's a function within a function. The chain rule provides a systematic way to find the derivative of such composite functions.
If we have a composite function y = f(g(x)), where f and g are differentiable functions, the chain rule states that the derivative of y with respect to x is the derivative of the outer function f evaluated at the inner function g(x), multiplied by the derivative of the inner function g(x) with respect to x. Mathematically, this can be expressed as:
dy/dx = f'(g(x)) * g'(x)
In essence, the chain rule tells us to differentiate the outer function, leaving the inner function untouched, and then multiply by the derivative of the inner function. This process allows us to unravel the layers of the composite function and find its derivative. For example, if we have y = (x^2 + 1)^3, the outer function is the cubing function, and the inner function is x^2 + 1. Applying the chain rule, we would first differentiate the outer function (cubing) to get 3*(x^2 + 1)^2, and then multiply by the derivative of the inner function (x^2 + 1), which is 2x. Thus, the derivative of the composite function is 3*(x^2 + 1)^2 * 2x.
The chain rule is indispensable for differentiating a wide range of functions, including trigonometric functions, exponential functions, and logarithmic functions. It is a fundamental tool in calculus and is essential for solving many real-world problems involving rates of change.
Applying the Chain Rule to Our Function
Now that we have a firm grasp of the power rule and the chain rule, let's apply these tools to find the derivative of our function, y = (3x^-3 + 2x)^3. This function is a composite function, where the outer function is the cubing function, and the inner function is the expression (3x^-3 + 2x). Therefore, we will need to employ the chain rule to differentiate it.
Following the chain rule, we first differentiate the outer function, which is the cubing function. The derivative of u^3 with respect to u is 3u^2. We evaluate this at the inner function, (3x^-3 + 2x), giving us:
3(3x^-3 + 2x)^2
Next, we need to multiply this by the derivative of the inner function, (3x^-3 + 2x), with respect to x. To find this derivative, we apply the power rule to each term.
The derivative of 3x^-3 is 3(-3)x^-4 = -9x^-4.
The derivative of 2x is 2.
Therefore, the derivative of the inner function (3x^-3 + 2x) is -9x^-4 + 2.
Finally, we multiply the derivative of the outer function by the derivative of the inner function:
dy/dx = 3(3x^-3 + 2x)^2 * (-9x^-4 + 2)
This is the derivative of our function y = (3x^-3 + 2x)^3 with respect to x. We have successfully applied the chain rule and the power rule to arrive at the solution.
Simplifying the Derivative (Optional)
The derivative we obtained in the previous section, dy/dx = 3(3x^-3 + 2x)^2 * (-9x^-4 + 2), is a valid answer. However, in some cases, it may be desirable to simplify the expression further. Simplification can make the derivative easier to work with and can also reveal underlying patterns or relationships.
To simplify our derivative, we can start by distributing the 3(3x^-3 + 2x)^2 term across the (-9x^-4 + 2) term:
dy/dx = 3(3x^-3 + 2x)^2 * (-9x^-4) + 3(3x^-3 + 2x)^2 * (2)
This gives us two terms. We can further simplify each term individually. Let's focus on the first term:
3(3x^-3 + 2x)^2 * (-9x^-4) = -27x^-4 (3x^-3 + 2x)^2
We can rewrite x^-4 as 1/x^4, which might make the expression look cleaner:
-27(1/x^4) (3x^-3 + 2x)^2
Now, let's look at the second term:
3(3x^-3 + 2x)^2 * (2) = 6(3x^-3 + 2x)^2
Combining the simplified terms, we get:
dy/dx = -27(1/x^4) (3x^-3 + 2x)^2 + 6(3x^-3 + 2x)^2
We can factor out the common term (3x^-3 + 2x)^2:
dy/dx = (3x^-3 + 2x)^2 [-27(1/x^4) + 6]
This is a simplified form of the derivative. Whether or not further simplification is necessary depends on the context of the problem and the desired level of detail. In some cases, the original form of the derivative might be perfectly acceptable, while in other cases, a simplified form might be more useful.
Common Mistakes to Avoid
When differentiating complex functions like y = (3x^-3 + 2x)^3, it's easy to make mistakes. Here are some common pitfalls to avoid:
- Forgetting the Chain Rule: The most common mistake is forgetting to apply the chain rule. Remember that when differentiating a composite function, you need to differentiate the outer function, leaving the inner function untouched, and then multiply by the derivative of the inner function.
- Incorrectly Applying the Power Rule: The power rule is simple, but it's crucial to apply it correctly. Make sure you multiply by the original exponent and reduce the exponent by 1.
- Errors in Arithmetic: Be careful with arithmetic, especially when dealing with negative exponents and fractions. A small arithmetic error can lead to a completely wrong answer.
- Not Simplifying (When Necessary): While simplification is not always required, it's important to know when it's necessary. If you need to use the derivative in further calculations, a simplified form will often make your work easier.
- Rushing the Process: Differentiation can be a multi-step process, especially for complex functions. Don't rush. Take your time, break the problem down into smaller steps, and double-check your work.
By being aware of these common mistakes, you can increase your accuracy and confidence in differentiating complex functions.
Conclusion
In this article, we have meticulously dissected the process of finding the derivative of y = (3x^-3 + 2x)^3 with respect to x. We have employed the fundamental rules of differentiation, namely the power rule and the chain rule, to arrive at the solution:
dy/dx = 3(3x^-3 + 2x)^2 * (-9x^-4 + 2)
We also explored optional simplification techniques and highlighted common mistakes to avoid during differentiation. The derivative, a cornerstone of calculus, reveals the instantaneous rate of change of a function. Mastering differentiation techniques empowers us to analyze the behavior of functions, solve optimization problems, and model real-world phenomena. Understanding the power rule and the chain rule, along with careful application and attention to detail, is key to success in calculus and related fields. This exploration serves as a testament to the power and elegance of calculus in unraveling the complexities of mathematical functions.
By grasping the concepts and techniques discussed in this article, you can confidently tackle similar differentiation problems and further expand your knowledge of calculus. The journey of learning calculus is an ongoing process, and each problem solved brings us closer to a deeper understanding of the mathematical world.
