How to Take the Derivative of a Reciprocal (the Reciprocal Rule)

Step-by-Step Guide

1. 1

   Step 1: Find the bottom (denominator).

   This is the only major factor in the derivative. ,, Note:
   Taking the negative derivative and dividing it by the bottom denominator squared might seem like a load of mumbo jumbo but it is dx/dy 1/f =
   -f'/f^2 where f' is the derivative of f.
   
   Examples:
   If a coefficient is there, multiply by the coefficient.
   
   For example, f (x) = 1/f , f'(x) =
   -f' / f^2.
   
   As a more complicated example, let's set f(x) = 3x^2.
   
   Then the reciprocal is 1/(3x^2).
   
   The negative derivative of the denominator is
   -2x because first we pull out the coefficient, 1/3;
   -2x becomes the new numerator.
   
   The denominator is squared, leaving us so far with dx/dy f(x) = 1/3 (-2x)/(x^2). , This can be simplified to 1/3 (-2/x^3), or
   -2/(3x^3).
2. 2

   Step 2: Take the negative derivative of the denominator; that becomes the numerator of the new derivative.

3. 3

   Step 3: Divide by the bottom denominator squared.

4. 4

   Step 4: Square the denominator to obtain x^4

5. 5

   Step 5: and the final result is f'(x) = dx/dy = 1/3 (-2x/x^4).

Detailed Guide

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