Question

the following expression occurs frequently in calculus. simplify the expression.
\frac{\frac{x^{2}}{sqrt{x^{2}+6}}-sqrt{x^{2}+6}}{x^{2}}

Explanation:

Step1: Combine the terms in the numerator

First, get a common - denominator for the terms in the numerator. The numerator is $\frac{x^{2}}{\sqrt{x^{2}+6}}-\sqrt{x^{2}+6}$. Rewrite $\sqrt{x^{2}+6}$ as $\frac{\sqrt{x^{2}+6}\cdot\sqrt{x^{2}+6}}{\sqrt{x^{2}+6}}=\frac{x^{2}+6}{\sqrt{x^{2}+6}}$. Then the numerator becomes $\frac{x^{2}-(x^{2}+6)}{\sqrt{x^{2}+6}}=\frac{x^{2}-x^{2}-6}{\sqrt{x^{2}+6}}=\frac{-6}{\sqrt{x^{2}+6}}$.

Step2: Divide by $x^{2}$

We have the original expression $\frac{\frac{x^{2}}{\sqrt{x^{2}+6}}-\sqrt{x^{2}+6}}{x^{2}}$. After simplifying the numerator to $\frac{-6}{\sqrt{x^{2}+6}}$, the expression is $\frac{\frac{-6}{\sqrt{x^{2}+6}}}{x^{2}}$. When dividing by $x^{2}$, we can write it as $\frac{-6}{x^{2}\sqrt{x^{2}+6}}$.

Answer:

$-\frac{6}{x^{2}\sqrt{x^{2}+6}}$