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How do you use the Quotient Rule to prove the Reciprocal Rule?

Answer 1

Charles Arocha

#d/(dx)(f(x)/g(x)) = (f'(x) g(x) - f(x) g'(x))/ (g(x))^2#

and #d/(dx)(1)=0#

#d/(dx)(1/g(x)) = (0* g(x) - 1* g'(x))/ (g(x))^2 = (- g'(x))/ (g(x))^2#

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Answer 2

Cameron Alexander

To prove the Reciprocal Rule using the Quotient Rule, consider the function ( f(x) = \frac{1}{g(x)} ), where ( g(x) ) is a differentiable function. Apply the Quotient Rule to ( f(x) ):

[ f'(x) = \frac{d}{dx} \left( \frac{1}{g(x)} \right) = \frac{g(x) \cdot 0 - 1 \cdot g'(x)}{(g(x))^2} ]

Simplify to obtain:

[ f'(x) = -\frac{g'(x)}{(g(x))^2} ]

This result shows that the derivative of ( f(x) ) is equal to the negative of the derivative of ( g(x) ) divided by the square of ( g(x) ), which is the Reciprocal Rule.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.