Understanding the nth Derivative of x3 log(x)
One of the fundamental tasks in calculus is to find the derivative of more complex functions. In this article, we will explore how to find the nth derivative of the function y x3 log(x). We will use the product rule and Leibniz’s rule to achieve this, presenting a detailed step-by-step guide to understanding the derivatives and the pattern that emerges.
Introduction to the Problem
The function y x3 log(x) is a product of two functions, u x3 and v log(x). Understanding the behavior of higher-order derivatives for such functions is crucial for a wide range of applications in mathematics, physics, and engineering.
Step 1: First Few Derivatives
Let's begin by finding the first few derivatives to identify any patterns.
First Derivative
Using the product rule (uv)' u'v uv', we find the first derivative of our function.
y x3 log(x)
Let u x3 and v log(x)
Then, u' 3x2 and v' 1/x
Therefore,
y' (3x2)(log(x)) (x3)(1/x)
Simplifying, we get
y' 3x2 log(x) x2
Second Derivative
Next, we take the derivative of the first derivative.
y' 3x2 log(x) x2
Using the product rule again, we get
y'' (6x)(log(x)) (3x2)(1/x) (2x)
Simplifying, we have
y'' 6x log(x) 3x 2x
Third Derivative
Now, let's find the third derivative.
y'' 6x log(x) 5x
Taking the derivative, we get
y''' 6(log(x)) 6x(1/x) 5
Simplifying, we obtain
y''' 6 log(x) 6 5
Step 2: General Derivative Expression
Patterns emerge when we continue to differentiate the function. We find that the general form of the nth derivative for n ≥ 3 can be expressed as:
yn Pn(x) log(x) Qn(x)
where Pn(x) and Qn(x) are polynomials in x with degrees n-3 and n-1, respectively.
Conclusion
While finding explicit forms for Pn(x) and Qn(x) for all n requires deeper combinatorial techniques or the use of Faà di Bruno's formula, the general structure of the nth derivative of y x3 log(x) can be summarized as:
yn Pn(x) log(x) Qn(x) for n ≥ 3
This structure provides a foundation for understanding and solving similar problems in calculus and related fields.