If I differentiate each of numerator, denominator, is result equal to orig fraction?

victoradri

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i know that if you perform the same operation to the numerator and the denominator of a fraction it still holds the same value. So , if i take the derivative of the numerator and the denominator , does the fraction still hold the same value?

Thanks for your help!
 
i know that if you perform the same operation to the numerator and the denominator of a fraction it still holds the same value. So , if i take the derivative of the numerator and the denominator , does the fraction still hold the same value?

Thanks for your help!
That statement, in general, holds true ONLY for multiplication (and division).
 
For example, what you say you "know" is not true for simple addition of fractions:
\(\displaystyle \frac{1}{3}+ \frac{1}{2}\) is NOT equal to \(\displaystyle \frac{1+ 1}{3+ 2}= \frac{2}{5}\). It is, rather, \(\displaystyle \frac{2}{6}+ \frac{3}{6}= \frac{5}{6}\).

The derivative of the fraction \(\displaystyle \frac{f(x)}{g(x)}\) is, using the "quotient rule", \(\displaystyle \frac{f'(x)g- fg'(x)}{g^2(x)}\)
 
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i know that if you perform the same operation to the numerator and the denominator of a fraction it still holds the same value. So , if i take the derivative of the numerator and the denominator , does the fraction still hold the same value?

Thanks for your help!

Try an example! Do it for, say, x^2/(x+1).

But your question doesn't really make sense, because a fraction of functions doesn't have a single value in the first place! If you mean, are f(x)/g(x) and f'(x)/g'(x) equivalent functions (with the same value for all x), the answer is no; for example, two functions with the same value for a given value of x typically have different derivatives at the same point.

But an example should make that clear.
 
i know that if you perform the same operation to the numerator and the denominator of a fraction it still holds the same value. So , if i take the derivative of the numerator and the denominator , does the fraction still hold the same value?

Thanks for your help!
So you think that \(\displaystyle \frac{2}{6}= \frac{2+5}{6+5}\) and \(\displaystyle \frac{5}{6}= \frac{5^2}{6^2}\). None of this is true. Your statement only holds for multiplication as already noted.
 
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