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Консультация № 177283
16.03.2010, 14:13
35.91 руб.
16.03.2010, 14:23
0
2
2
Здравствуйте, уважаемые эксперты!
Найти производные данных функций, используя приём логарифмического дифференцирования.
Найти производные данных функций, используя приём логарифмического дифференцирования.
Обсуждение
16.03.2010, 15:15
общий
это ответ
Здравствуйте, sanekvseti.
1. y = (4x + 5)^(cos 4x),
ln y = (cos 4x)*ln(4x + 5),
(ln y)’ = ((cos 4x)*ln(4x + 5))’,
y’/y = (cos 4x)’*ln(4x + 5) + (cos 4x)*(ln (4x + 5))’ = -4*(sin 4x)*ln(4x + 5) + 4*(cos 4x)*1/(4x + 5),
y’ = y*(-4*(sin 4x)*ln(4x + 5) + 4*(cos 4x)*1/(4x + 5)) =
= 4*((4x + 5)^(cos 4x))*((cos 4x)/(4x + 5) – (sin 4x)*ln (4x + 5)).
2. y = (e^(3x + 1))*((2x^2 – 5x + 7)^10)/((8x + 7)^(2/3)),
ln y = (3x + 1) + 10*ln(2x^2 – 5x + 7) – (2/3)*ln(8x + 7),
(ln y)’ = ((3x + 1) + 10*ln(2x^2 – 5x + 7) – (2/3)*ln(8x + 7))’,
y’/y = 3 + 10*(1/(2x^2 – 5x + 7))*(4x – 5) – (2/3)*(1/(8x + 7))*8,
y’ = y*(3 + 10*(1/(2x^2 – 5x + 7))*(4x – 5) – (2/3)*(1/(8x + 7))*8) =
= (e^(3x + 1))*((2x^2 – 5x + 7)^10)/((8x + 7)^(2/3))*(3 + 10*(4x – 5)/(2x^2 – 5x + 7) – (16/3)*(1/(8x + 7))).
С уважением.
1. y = (4x + 5)^(cos 4x),
ln y = (cos 4x)*ln(4x + 5),
(ln y)’ = ((cos 4x)*ln(4x + 5))’,
y’/y = (cos 4x)’*ln(4x + 5) + (cos 4x)*(ln (4x + 5))’ = -4*(sin 4x)*ln(4x + 5) + 4*(cos 4x)*1/(4x + 5),
y’ = y*(-4*(sin 4x)*ln(4x + 5) + 4*(cos 4x)*1/(4x + 5)) =
= 4*((4x + 5)^(cos 4x))*((cos 4x)/(4x + 5) – (sin 4x)*ln (4x + 5)).
2. y = (e^(3x + 1))*((2x^2 – 5x + 7)^10)/((8x + 7)^(2/3)),
ln y = (3x + 1) + 10*ln(2x^2 – 5x + 7) – (2/3)*ln(8x + 7),
(ln y)’ = ((3x + 1) + 10*ln(2x^2 – 5x + 7) – (2/3)*ln(8x + 7))’,
y’/y = 3 + 10*(1/(2x^2 – 5x + 7))*(4x – 5) – (2/3)*(1/(8x + 7))*8,
y’ = y*(3 + 10*(1/(2x^2 – 5x + 7))*(4x – 5) – (2/3)*(1/(8x + 7))*8) =
= (e^(3x + 1))*((2x^2 – 5x + 7)^10)/((8x + 7)^(2/3))*(3 + 10*(4x – 5)/(2x^2 – 5x + 7) – (16/3)*(1/(8x + 7))).
С уважением.
Об авторе:
Facta loquuntur.
Facta loquuntur.
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