SOLUTION: Using Limits to Calculate a Derivative Use the definition of the derivative to find f ′(x) if f(x)= 1/(x-1)

Algebra ->  Equations -> SOLUTION: Using Limits to Calculate a Derivative Use the definition of the derivative to find f ′(x) if f(x)= 1/(x-1)       Log On


   

Question 1010141: Using Limits to Calculate a Derivative
Use the definition of the derivative to find f ′(x) if
f(x)= 1/(x-1)

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
f%28x%29+=+1%2F%28x-1%29


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Use f(x) to find f(x+h)


f%28x%29+=+1%2F%28x-1%29


f%28x%2Bh%29+=+1%2F%28x%2Bh-1%29


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Compute the difference quotient


%28f%28x%2Bh%29-f%28x%29%29%2Fh+=+%281%2F%28x%2Bh-1%29-1%2F%28x-1%29%29%2Fh














%28f%28x%2Bh%29-f%28x%29%29%2Fh+=+%28-h%29%2F%28h%28x-1%29%28x%2Bh-1%29%29





%28f%28x%2Bh%29-f%28x%29%29%2Fh+=+%28-1%29%2F%28%28x-1%29%28x%2Bh-1%29%29


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The difference quotient is %28f%28x%2Bh%29-f%28x%29%29%2Fh+=+%28-1%29%2F%28%28x-1%29%28x%2Bh-1%29%29


Now evaluate the limit as h goes to 0. Since we no longer have to worry about division by zero errors, this is the equivalent to plugging in h = 0 to get





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So the derivative of f(x) is f ' (x) = +%28-1%29%2F%28%28x-1%29%5E2%29