Use the Limit Definition to Find the Derivative f(x)=1/(x-1)

Problem

ƒ(x)=1/(x−1)

Solution

1. State the limit definition of the derivative, which is the limit of the difference quotient as h approaches zero.

ƒ(x)′=(lim_h→0)((ƒ*(x+h)−ƒ(x))/h)

2. Substitute the function into the definition by replacing ƒ*(x+h) with 1/((x+h)−1) and ƒ(x) with 1/(x−1)

ƒ(x)′=(lim_h→0)((1/(x+h−1)−1/(x−1))/h)

3. Find a common denominator for the fractions in the numerator to combine them.

ƒ(x)′=(lim_h→0)(((x−1)−(x+h−1))/((x+h−1)*(x−1))/h)

4. Simplify the numerator by distributing the negative sign and canceling terms.

ƒ(x)′=(lim_h→0)((x−1−x−h+1)/((x+h−1)*(x−1))/h)

ƒ(x)′=(lim_h→0)((−h)/((x+h−1)*(x−1))/h)

5. Divide by h by multiplying the numerator fraction by the reciprocal of h

ƒ(x)′=(lim_h→0)((−h)/(h*(x+h−1)*(x−1)))

6. Cancel the common factor of h from the numerator and denominator.

ƒ(x)′=(lim_h→0)((−1)/((x+h−1)*(x−1)))

7. Evaluate the limit by substituting h=0 into the expression.

ƒ(x)′=(−1)/((x+0−1)*(x−1))

ƒ(x)′=(−1)/((x−1)*(x−1))

Final Answer

d()/d(x)1/(x−1)=−1/((x−1)2)

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