"Введение в матанализ", Производная

Задача 32

Условие

Найти производную: y equals the log base x of open paren 2 to the x-th power close paren equals f with hook of x

Решение

f with hook of x equals the fraction with numerator l n of open paren 2 to the x-th power close paren and denominator l n of x equals x l n of 2 over l n of x equals x over the log base 2 of x

f with hook prime of x equals the fraction with numerator the log base 2 of x minus open paren the log base 2 of x close paren prime times x and denominator log base 2 squared x equals the fraction with numerator the log base 2 of x minus 1 over l n of 2 x x and denominator log base 2 squared x equals the fraction with numerator l n of x over l n of 2 minus 1 over l n of 2 and denominator l n of x over l n of 2 times the log base 2 of x equals the fraction with numerator l n of x minus 1 and denominator l n of x times the log base 2 of x

Ответ: the fraction with numerator l n of x minus 1 and denominator l n of x times the log base 2 of x comma x is greater than 0 comma x is not equal to 1

Задача 74

Условие

Найти производную: y equals the square root of 1 plus the tangent of open paren x squared plus x to the negative 2 power close paren equals f with hook of x

Решение

f with hook prime of x equals one half times open paren 1 plus the tangent of open paren x squared plus x to the negative 2 power close paren close paren raised to the exponent negative one half end exponent times open paren 1 plus the tangent of open paren x squared plus x to the negative 2 power close paren close paren prime equals

equals 1 over 2 the square root of 1 plus the tangent of open paren x squared plus x to the negative 2 power close paren times the tangent prime of open paren x squared plus x to the negative 2 power close paren times open paren x squared plus x to the negative 2 power close paren prime equals

equals 1 over 2 the square root of 1 plus the tangent of open paren x squared plus x to the negative 2 power close paren times the cosine squared of open paren x squared plus x to the negative 2 power close paren times 2 times open paren x minus the fraction with numerator 1 and denominator x cubed close paren equals

equals the fraction with numerator x to the fourth power minus 1 and denominator x cubed the square root of 1 plus the tangent of open paren x squared plus x to the negative 2 power close paren times the cosine of open paren x squared plus x to the negative 2 power close paren

Ответ: the fraction with numerator x to the fourth power minus 1 and denominator x cubed the square root of 1 plus the tangent of open paren x squared plus x to the negative 2 power close paren times the cosine of open paren x squared plus x to the negative 2 power close paren

Задача 117

Условие

Найти производную: y equals the fraction with numerator x and denominator the square root of ℇ raised to the 2 x power minus 1 minus the arc tangent of open paren the square root of ℇ raised to the 2 x power minus 1 close paren equals f with hook of x

Решение

open paren ℇ raised to the 2 x power minus 1 close paren prime equals open paren ℇ raised to the 2 x power close paren prime equals 2 ℇ raised to the 2 x power

open paren the square root of ℇ raised to the 2 x power minus 1 close paren prime equals the fraction with numerator ℇ raised to the 2 x power and denominator the square root of ℇ raised to the 2 x power minus 1

open paren the fraction with numerator x and denominator the square root of ℇ raised to the 2 x power minus 1 close paren prime equals the fraction with numerator the square root of ℇ raised to the 2 x power minus 1 minus the fraction with numerator ℇ raised to the 2 x power and denominator the square root of ℇ raised to the 2 x power minus 1 x and denominator ℇ raised to the 2 x power minus 1

open paren the arc tangent of open paren the square root of ℇ raised to the 2 x power minus 1 close paren close paren prime equals the fraction with numerator 1 and denominator 1 plus ℇ raised to the 2 x power minus 1 times the fraction with numerator ℇ raised to the 2 x power and denominator the square root of ℇ raised to the 2 x power minus 1 equals the fraction with numerator 1 and denominator the square root of ℇ raised to the 2 x power minus 1

f with hook prime of x equals the fraction with numerator negative the fraction with numerator ℇ raised to the 2 x power and denominator the square root of ℇ raised to the 2 x power minus 1 x and denominator ℇ raised to the 2 x power minus 1 equals the fraction with numerator negative x ℇ raised to the 2 x power and denominator the square root of open paren ℇ raised to the 2 x power minus 1 close paren cubed

Ответ: the fraction with numerator negative x ℇ raised to the 2 x power and denominator the square root of open paren ℇ raised to the 2 x power minus 1 close paren cubed

Задача 149

Условие

Найти производную: y equals x raised to the exponent x to the x-th power end exponent equals f with hook of x

Решение

x to the x-th power prime equals x to the x-th power times open paren l n of x times x close paren prime equals x to the x-th power times open paren 1 plus l n of x close paren

f with hook prime of x equals x raised to the exponent x to the x-th power end exponent prime equals x raised to the exponent x to the x-th power end exponent times open paren l n of x times x to the x-th power close paren prime equals x raised to the exponent x to the x-th power end exponent times open paren x raised to the x minus 1 power plus l n of x times x to the x-th power prime close paren equals

equals x raised to the exponent x to the x-th power end exponent times open paren x raised to the x minus 1 power plus l n of x times x to the x-th power times open paren l n of x plus 1 close paren close paren

Ответ: x raised to the exponent x to the x-th power end exponent times open paren x raised to the x minus 1 power plus l n of x times x to the x-th power times open paren l n of x plus 1 close paren close paren

Теорема 3.

Условие

Найти производную, не упрощая: y equals open paren the fraction with numerator the arc sine of open paren the square root of x close paren plus the cosine squared of open paren 3 x minus 1 close paren and denominator 5 raised to the x cubed power plus l n squared of open paren 1 plus ℇ to the x-th power close paren close paren raised to the exponent x squared hyperbolic cosine of x end exponent

Решение

(Здесь придется листать вправо, прошу прощения)y prime equals open paren the fraction with numerator the arc sine of open paren the square root of x close paren plus the cosine squared of open paren 3 x minus 1 close paren and denominator 5 raised to the x cubed power plus l n squared of open paren 1 plus ℇ to the x-th power close paren close paren raised to the exponent x squared hyperbolic cosine of x end exponent times open paren l n of open paren the fraction with numerator the arc sine of open paren the square root of x close paren plus the cosine squared of open paren 3 x minus 1 close paren and denominator 5 raised to the x cubed power plus l n squared of open paren 1 plus ℇ to the x-th power close paren close paren times x squared hyperbolic cosine of x close paren prime equals

equals open paren the fraction with numerator the arc sine of open paren the square root of x close paren plus the cosine squared of open paren 3 x minus 1 close paren and denominator 5 raised to the x cubed power plus l n squared of open paren 1 plus ℇ to the x-th power close paren close paren raised to the exponent x squared hyperbolic cosine of x end exponent times open paren l n of open paren the fraction with numerator the arc sine of open paren the square root of x close paren plus the cosine squared of open paren 3 x minus 1 close paren and denominator 5 raised to the x cubed power plus l n squared of open paren 1 plus ℇ to the x-th power close paren close paren times open paren 2 x hyperbolic cosine of x plus x squared hyperbolic sine of x close paren plus x squared hyperbolic cosine of x times the fraction with numerator 5 raised to the x cubed power plus l n squared of open paren 1 plus ℇ to the x-th power close paren and denominator the arc sine of open paren the square root of x close paren plus the cosine squared of open paren 3 x minus 1 close paren times the fraction with numerator open paren the arc sine of open paren the square root of x close paren plus the cosine squared of open paren 3 x minus 1 close paren close paren times open paren 5 raised to the x cubed power times open paren 3 times l n of 5 close paren plus 2 times l n of open paren open paren 1 plus ℇ to the x-th power close paren close paren times the fraction with numerator 1 and denominator 1 plus ℇ to the x-th power times ℇ to the x-th power close paren minus open paren 5 raised to the x cubed power plus l n squared of open paren 1 plus ℇ to the x-th power close paren close paren times open paren the fraction with numerator 1 and denominator the square root of 1 minus x times the fraction with numerator 1 and denominator the square root of x plus 2 the cosine of open paren 3 x minus 1 close paren times negative the sine of open paren 3 x minus 1 close paren times 3 close paren and denominator open paren 5 raised to the x cubed power plus l n squared of open paren 1 plus ℇ to the x-th power close paren close paren squared close paren