Resolved: "How do you use the formal definition of..."

George Michael

Answered question

2022-04-09

How do you use the formal definition of differentiation as a limit to find the derivative of

f (x) = \frac{1}{x - 1}

firyemv3

Beginner2022-04-10Added 10 answers

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

= lim_{h \to 0} \frac{1}{h} \frac{x - 1 - x - h + 1}{[(x + h) - 1] (x - 1)} = h \to 0 \frac{- 1}{(x - 1) (x - 1)} = - \frac{1}{{(x - 1)}^{2}}

analiticozuod

Beginner2022-04-11Added 10 answers

Firstly, let's remember the limit definition of the derivative:

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

Here, we have the function

f (x) = \frac{1}{x - 1}

, so:

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

= lim_{h \to 0} \frac{(x + 1) - (x + h + 1)}{h (x + h + 1) (x + 1)}

= lim_{h \to 0} \frac{(- h)}{h (x + h + 1) (x + 1)}

= lim_{h \to 0} - \frac{1}{(x + h + 1) (x + 1)}

= - \frac{1}{(x + 0 + 1) (x + 1)} = - \frac{1}{{(x + 1)}^{2}}

That's it.

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