Simplify each complex fraction. [ 1/[(x+h)^2 + 9] - 1/(x^2+9)] / h

Verified step by step guidance

1. Identify the complex fraction: $\frac{\frac{1}{(x+h{)}^2+9}-\frac{1}{x^2+9}}{h}$.
2. Recognize that the numerator is a difference of two fractions. Find a common denominator for these fractions, which is $(x+h{)}^2+9)(x^2+9)$.
3. Rewrite each fraction in the numerator with the common denominator: $\frac{x^2+9}{((x+h{)}^2+9)(x^2+9)}-\frac{(x+h{)}^2+9}{((x+h{)}^2+9)(x^2+9)}$.
4. Combine the fractions in the numerator: $\frac{x^2+9-((x+h{)}^2+9)}{((x+h{)}^2+9)(x^2+9)}$.
5. Simplify the expression in the numerator and then divide by $h$ by multiplying by its reciprocal: $\frac{x^2+9-(x^2+2xh+h^2+9)}{h((x+h{)}^2+9)(x^2+9)}$.

Key Concepts

Complex Fractions

A complex fraction is a fraction where the numerator, the denominator, or both contain fractions themselves. To simplify complex fractions, one typically finds a common denominator for the fractions involved, allowing for easier manipulation and simplification of the overall expression.

Limit Definition

In calculus, the limit definition of a derivative involves the concept of approaching a value as a variable approaches zero. In the context of the given expression, as 'h' approaches zero, the simplification of the complex fraction can reveal the derivative of a function, which is foundational in understanding rates of change.

Algebraic Manipulation

Algebraic manipulation involves applying various algebraic techniques, such as factoring, distributing, and combining like terms, to simplify expressions. In the context of the given problem, manipulating the complex fraction will require careful handling of the terms to achieve a simplified form that can be further analyzed or evaluated.