Instruction

1

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.2

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.3

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).

4

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).

5

Example 2[(x? — 1) / 6x]’ = [(2x · 6x — 6 · x?) / 6?] = [12x? — 6x?] / 36 = 6x? / 36 = x? / 6.

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).7

Example(1 / x)' = [x^(-1)]' = -1 · x^(-2) = -1 / x?.

Note

The fraction may contain in its composition of several fractions. In this case, it is easier to find first separately derived "primary" fractions.

Useful advice

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.