Any fraction has a numerator and denominator. In the process of finding the derivative of a fraction need separately finding the derivative of the numerator and the derivative of the denominator.
To find the derivative from the fraction, the derivative of the numerator to the denominator comnote. Subtract from the resulting expression of the derivative of the denominator multiplied by the numerator. The result, divide by the denominator squared.
Пример 1[sin (x) / cos (x)]’ = [sin (x) · cos (x) — cos’ (x) · sin (x)] / cos? (x) = [cos (x) · cos (x) + sin (x) · sin (x)] / cos? (x) = [cos? (x) + sin? (x)] / cos? (x) = 1 / cos? (x).
The result is nothing but a table-valued derivative of the function tangent. It is understandable, because the ratio of the sine to the cosine is, by definition, the tangent. Thus,tg (x) = [sin (x) / cos (x)]' = 1 / cos? (x).
Example 2[(x? — 1) / 6x]’ = [(2x · 6x — 6 · x?) / 6?] = [12x? — 6x?] / 36 = 6x? / 36 = x? / 6.
A special case of the fraction is a fraction whose denominator unit. Find the derivative of such type of fractions you just have to present it in the form of a denominator with degree (-1).
Example(1 / x)' = [x^(-1)]' = -1 · x^(-2) = -1 / x?.
The fraction may contain in its composition of several fractions. In this case, it is easier to find first separately derived "primary" fractions.
When you are looking for the derivative of the denominator and the numerator, apply the rules of differentiation: sum, product, complex functions. It is useful to keep in mind the simplest derivatives of tabular functions: linear, exponential, power, logarithmic, trigonometric, etc.