How can you derive the quotient rule?

Answer 1
This can be proven fairly quickly, assuming knowledge of prior subjects such as the product rule and chain rule. Suppose #f(x) = (u(x))/(v(x))#. As we know that all of our equations are in terms of #x#, henceforth #x# will be omitted from the steps below. Note however that it is still present as the variable for the functions.
#(d/dx)f = (d/dx)u/v#
Then via our definition #f= u/v# we get #u= f*v#. Differentiating this via use of the product rule nets us...
#u' = f'*v + f*v'#

Now as we isolate f' on its own side...

#f'= [u'-f*v']/(v)#
Recalling that #f=u/v# this becomes...
#f' = [u' - (u/v)*v']/v#
And by multiplying both the numerator and denominator by #v# we get...
#f' = [u'*v - u*v']/[v^2]#
Or, by showing #x# again...
#f'(x) = [u'(x)*v(x) - u(x)*v'(x)]/(v(x))^2#
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Answer 2

To derive the quotient rule, consider two functions, u(x) and v(x), and their respective derivatives u'(x) and v'(x). The quotient rule states that the derivative of the quotient of two functions, u(x)/v(x), is given by:

(d/dx)[u(x)/v(x)] = (v(x)*u'(x) - u(x)*v'(x)) / [v(x)]^2

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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.

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.

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.

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.

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