Question

the following expression occurs frequently in calculus. simplify the expression.
\\(\frac{\sqrt{x}-\frac{1}{10\sqrt{x}}}{\sqrt{x}}\\)

Explanation:

Step1: Combine terms in numerator

First, find a common - denominator for the terms in the numerator. The common denominator of $\sqrt{x}$ and $\frac{1}{10\sqrt{x}}$ is $10\sqrt{x}$. So, $\sqrt{x}=\frac{10x}{10\sqrt{x}}$. Then $\sqrt{x}-\frac{1}{10\sqrt{x}}=\frac{10x - 1}{10\sqrt{x}}$.

Step2: Divide by $\sqrt{x}$

We have the expression $\frac{\frac{10x - 1}{10\sqrt{x}}}{\sqrt{x}}$. When dividing by a number is the same as multiplying by its reciprocal, so $\frac{\frac{10x - 1}{10\sqrt{x}}}{\sqrt{x}}=\frac{10x - 1}{10\sqrt{x}}\times\frac{1}{\sqrt{x}}$.

Step3: Simplify the product

Multiply the fractions: $\frac{10x - 1}{10\sqrt{x}}\times\frac{1}{\sqrt{x}}=\frac{10x - 1}{10x}$.

Answer:

$\frac{10x - 1}{10x}$