The Derivative of y x^2 - 5x6^4 Explained
In calculus, the concept of a derivative is fundamental. It measures the rate at which a function is changing. This article will delve into finding the derivative of the polynomial function y x2 - 5x64. We will explore the step-by-step process, explaining each part of the differentiation, and the application of the chain rule.
Understanding Polynomial Functions and Their Derivatives
A polynomial function is a function consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. The derivative of a polynomial function is another polynomial whose degree is one less than the original function. For instance, if we have a polynomial of the form axn, its derivative is anxn-1.
The Problem: Differentiating y x2 - 5x64
The given function is y x2 - 5x64, and we need to find its derivative with respect to x.
Step 1: Identify Each Term
The function consists of two terms: x2 and -5x64. To differentiate, we will apply the power rule to each term individually.
Step 2: Applying the Power Rule
The power rule states that for any function of the form xn, the derivative is nxn-1. Let's apply this rule to each term:
x2: The derivative is 2x2-1 2x. -5x64: First, simplify the term to -5x64. The derivative of x6 is 6x5. Therefore, the derivative of -5x64 is -5 times; 6x5 -35.Step 3: Combining the Terms
Now, we combine the derivatives of each term:
[frac{dy}{dx} 2x - 3^5]The Role of the Chain Rule
In this particular problem, although it is simple and does not require the chain rule explicitly, it is worth mentioning the chain rule. The chain rule is used when we have a composite function, where one function is nested within another. For example, if we had y (x64)2, we would use the chain rule as follows:
[frac{dy}{dx} 2(x6^4) times 4x6^3 times 6]Simplifying further:
[frac{dy}{dx} 48x6^7]However, in our original problem, the function is not a nested form, so the chain rule is not directly needed.
Final Answer
The derivative of y x2 - 5x64 with respect to x is:
[frac{dy}{dx} 2x - 3^5]Conclusion
Understanding derivatives is crucial in calculus. By breaking down the problem and applying the power rule, we can differentiate polynomial functions efficiently. The chain rule, while not always needed, is an essential tool for more complex functions. If you are interested in exploring more advanced topics in calculus, such as higher-order derivatives and implicit differentiation, there are numerous resources available online and in textbooks.