Use the definition of derivative to find 𝑓′𝑥, where f( x)= 1/√(x-1)

To find the derivative of \( f(x) = \frac{1}{\sqrt{x-1}} \) using the definition of the derivative, we start with the formula:

\[
f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
\]

Given \( f(x) = \frac{1}{\sqrt{x-1}} \), we need to calculate \( f(x+h) \):

\[
f(x+h) = \frac{1}{\sqrt{(x+h)-1}} = \frac{1}{\sqrt{x+h-1}}
\]

Now, substitute \( f(x) \) and \( f(x+h) \) into the derivative formula:

\[
f'(x) = \lim_{h \to 0} \frac{\frac{1}{\sqrt{x+h-1}} - \frac{1}{\sqrt{x-1}}}{h}
\]

To simplify this expression, we will find a common denominator for the numerator:

\[
f'(x) = \lim_{h \to 0} \frac{\sqrt{x-1} - \sqrt{x+h-1}}{h \sqrt{x-1} \sqrt{x+h-1}}
\]

Next, we rationalize the numerator by multiplying the numerator and denominator by the conjugate:

\[
f'(x) = \lim_{h \to 0} \frac{(\sqrt{x-1} - \sqrt{x+h-1}) (\sqrt{x-1} + \sqrt{x+h-1})}{h \sqrt{x-1} \sqrt{x+h-1} (\sqrt{x-1} + \sqrt{x+h-1})}
\]

This simplifies to:

\[
f'(x) = \lim_{h \to 0} \frac{(x-1) - (x+h-1)}{h \sqrt{x-1} \sqrt{x+h-1} (\sqrt{x-1} + \sqrt{x+h-1})}
\]

\[
f'(x) = \lim_{h \to 0} \frac{-h}{h \sqrt{x-1} \sqrt{x+h-1} (\sqrt{x-1} + \sqrt{x+h-1})}
\]

Canceling \( h \) from the numerator and denominator gives:

\[
f'(x) = \lim_{h \to 0} \frac{-1}{\sqrt{x-1} \sqrt{x+h-1} (\sqrt{x-1} + \sqrt{x+h-1})}
\]

As \( h \to 0 \), \( \sqrt{x+h-1} \to \sqrt{x-1} \). Thus, the expression simplifies to:

\[
f'(x) = \frac{-1}{\sqrt{x-1} \cdot \sqrt{x-1} \cdot (2\sqrt{x-1})}
\]

\[
f'(x) = \frac{-1}{2 (x-1)^{3/2}}
\]

Therefore, the derivative of \( f(x) = \frac{1}{\sqrt{x-1}} \) is:

\[
f'(x) = -\frac{1}{2 (x-1)^{3/2}}
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you use the definition of derivative for trigonometric functions?
2. What is the derivative of \( f(x) = \sqrt{x+2} \) using the definition?
3. How can you verify this derivative using standard derivative rules?
4. What happens to the derivative at points where the function is undefined?
5. How does the derivative change if the function is \( f(x) = 1/\sqrt{x+1} \)?

### Tip:
When using the definition of a derivative, rationalizing the numerator or denominator can simplify the limit calculation.

Use the definition of derivative to find 𝑓′(x), where f(x) = 1/√(x-1)

Learn how to find the derivative of the function f(x) = 1/√(x-1) using the limit definition of the derivative. This detailed solution involves algebraic manipulation, rationalizing the numerator, and applying limits to find f'(x). Suitable for students in Grades 11-12 studying calculus.

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