Calculating the nth Order Derivative of 1/(x-x^2) For SEO

In this article, we will delve into the method for finding the nth order derivative of the function f(x) frac{1}{x-x^2}. This is a valuable skill for anyone studying calculus, particularly for optimizing Google searches and content for search engine optimization (SEO).

Introduction to the Function

We start with the function f(x) frac{1}{x-x^2}. This function can be rewritten and simplified for easier differentiation.

Step 1: Simplify the Function

First, let's rewrite the function in a more simplified form:
f(x) frac{1}{x(1-x)}
Using partial fractions, we can further simplify the function:

f(x) frac{1}{x(1-x)} frac{1}{x} - frac{1}{1-x}

Step 2: Differentiation

Now, let's proceed with differentiating the function with respect to x:

First derivative:

[ f^{(1)}(x) -frac{1}{x^2} - frac{1}{(1-x)^2} ]

Second derivative:

[ f^{(2)}(x) frac{2!}{x^3} - frac{2!}{(1-x)^3} ]

Third derivative:

[ f^{(3)}(x) -frac{3!}{x^4} - frac{3!}{(1-x)^4} ]

General pattern for nth derivative:

[ f^{(n)}(x) -frac{n!}{x^{n 1}} - frac{n!}{(1-x)^{n 1}} ]

Understanding the Pattern

The process of finding the nth order derivative of the function is based on observing the pattern in the derivatives we have already calculated. From the first few derivatives, we can generalize the form of the nth derivative.

Final Expression

Using the derived pattern, we can express the nth order derivative of the function f(x) frac{1}{x(1-x)} as follows:

[ f^{(n)}(x) -frac{n!}{x^{n 1}} - frac{n!}{(1-x)^{n 1}} -n!left(frac{1}{x^{n 1}} - frac{1}{(1-x)^{n 1}}right) ]

SEO Keyword Optimization

When optimizing this content for SEO, it is important to include relevant keywords that reflect the topic at hand. For keyword optimization:

Keyword: nth order derivative

Keyword: calculus

Keyword: higher order derivatives

Conclusion

By understanding and applying the method for finding the nth order derivative of the function, you can enhance your skills in calculus and improve your SEO efforts by creating content that is search-friendly and rich in meaningful keywords.