Provide an example on how to differentiate a fraction, if the maxim power of variable from numerator is smaller than denominator.

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example function : f(x)=(3x+14)/(3x^2+7x)

Since the given function is a quotient, we'll apply the quotient rule to find it's first derivative:

(u/v)' = (u'*v - u*v')/v^2 (*)

We'll put u = 3x+14 => u' = 3

We'll put v = 3x^2 + 7x => v' = 6x + 7

We'll substitute u,v,u',v' in the formula (*):

f'(x) = [3(3x^2 + 7x) - (3x+14)(6x + 7)]/(3x^2+7x)^2

We'll remove the brackets:

f'(x) = (9x^2 + 21x - 18x^2 - 21x - 84x - 98)/(3x^2+7x)^2

We'll combine and eliminate like terms:

**f'(x) = -(9x^2 + 84x + 98)/(3x^2+7x)^2**

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