Simplify the expression: $\frac{\sqrt{x+h+1} - \sqrt{x+1}}{h}$
Understand the Problem
The question presents a mathematical expression involving square roots and the variable h. It appears to be part of a limit definition of a derivative. The expression needs simplification or evaluation, likely involving rationalizing the numerator.
Answer
$\frac{1}{\sqrt{x+h+1} + \sqrt{x+1}}$
Answer for screen readers
$\frac{1}{\sqrt{x+h+1} + \sqrt{x+1}}$
Steps to Solve
- Multiply by the conjugate
To rationalize the numerator, we multiply the expression by the conjugate of the numerator, which is $\sqrt{x+h+1} + \sqrt{x+1}$, divided by itself. This doesn't change the value of the expression.
$$ \frac{\sqrt{x+h+1} - \sqrt{x+1}}{h} \cdot \frac{\sqrt{x+h+1} + \sqrt{x+1}}{\sqrt{x+h+1} + \sqrt{x+1}} $$
- Simplify the numerator
The numerator simplifies to a difference of squares: $(\sqrt{x+h+1})^2 - (\sqrt{x+1})^2 = (x+h+1) - (x+1)$.
$$ \frac{(x+h+1) - (x+1)}{h(\sqrt{x+h+1} + \sqrt{x+1})} $$
- Further simplify the numerator
Simplify the numerator by canceling terms: $x + h + 1 - x - 1 = h$.
$$ \frac{h}{h(\sqrt{x+h+1} + \sqrt{x+1})} $$
- Cancel the $h$ term
Cancel the $h$ in the numerator and the denominator.
$$ \frac{1}{\sqrt{x+h+1} + \sqrt{x+1}} $$
$\frac{1}{\sqrt{x+h+1} + \sqrt{x+1}}$
More Information
This expression is often encountered when finding the derivative of $\sqrt{x+1}$ using the limit definition. It represents the simplified form after rationalizing the numerator.
Tips
A common mistake is not correctly multiplying by the conjugate, or making errors when expanding and simplifying the numerator. Also, forgetting to keep the denominator when multiplying by the conjugate is another common mistake.
AI-generated content may contain errors. Please verify critical information
