Question

Вычислите производные сложных функций

Answer 1

а)

$f'(x) = 5(3 - 2 {x}^{3} ) {}^{4} \times (3 - 2 {x}^{3} )' = \\ = 5(3 - 2 {x}^{3} ) {}^{4} \times ( - 6 {x}^{2} ) = - 30 {x}^{2} (3 - 2 {x}^{3} ) {}^{4}$

б)

$f'(x) = ln(0.3) \times {0.3}^{3 {x}^{2} - 7x + 2} \times (3 {x}^{2} - 7x + 2) '= \\ = ln( 0.3 ) \times ( {0.3)}^{3 {x}^{2} - 7x + 2} \times (6x - 7)$

в)

$f'(x) = - \sin( {x}^{2} + 4x + 12) \times ( {x}^{2} + 4x + 12) '= \\ = - (2x + 4) \sin( {x}^{2} + 4x + 12 ) = \\ = ( - 2x - 4) \sin( {x}^{2} + 4x + 12 )$

г)

$f'(x) = (3x + 5 {x}^{2} + {x}^{3} )' \times {4}^{ {x}^{2} } + ( {4}^{ {x}^{2} } ) '\times ( {x}^{2} ) '\times ( 3x + 5 {x}^{2} + {x}^{3} ) = \\ = (3 + 10x + 3 {x}^{2} ) \times {4}^{ {x}^{2} } + ln(4) \times {4}^{ {x}^{2} } \times 2x(3x + 5 {x}^{2} + {x}^{3} ) = \\ = {4}^{ {x}^{2} } ((3 + 10x + 3 {x}^{2} ) + (6 {x}^{2} + 10 {x}^{3} + 2 {x}^{4} ) ln(4))$

д)

$f'(x) = 3 \times 2 \sin(5x) \times (\sin(5x)) ' \times (5x) '= \\ = 6 \sin(5x) \times \cos(5x) \times 5 = 15 \sin(10x)$