The Second Derivative

Calculus Level 2

Find the second derivative of f ( x ) = 4 ( 1 + 3 x ) 5 f(x) = 4(1+3x)^5 at x = 1 3 x = \dfrac13 .


The answer is 5760.

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2 solutions

f ( x ) = 4 ( 1 + 3 x ) 5 f ( x ) = 4 5 ( 1 + 3 x ) 5 1 3 f ( x ) = 60 ( 1 + 3 x ) 4 f ( x ) = 60 4 ( 1 + 3 x ) 3 3 f ( x ) = 720 ( 1 + 3 x ) 3 f ( 1 3 ) = 720 ( 1 + 3 1 3 ) 3 f ( 1 3 ) = 720 ( 1 + 3 3 ) 3 f ( 1 3 ) = 720 ( 2 ) 3 f ( 1 3 ) = 720 8 f ( 1 3 ) = 5760 f(x)\quad =\quad 4*{ (1+3x })^{ 5 }\\ \\ f'(x)\quad =\quad 4*5*{ (1+3x })^{ 5-1 }*3\\ f'(x)\quad =\quad 60*{ (1+3x) }^{ 4 }\\ \\ f''(x)\quad =\quad 60*4*{ (1+3x) }^{ 3 }*3\\ f''(x)\quad =\quad 720*{ (1+3x })^{ 3 }\\ \\ f''(\frac { 1 }{ 3 } )\quad =\quad 720*{ (1+3*\frac { 1 }{ 3 } ) }^{ 3 }\\ f''(\frac { 1 }{ 3 } )\quad =\quad 720*{ (1+\frac { 3 }{ 3 } ) }^{ 3 }\\ f''(\frac { 1 }{ 3 } )\quad =\quad 720*{ (2) }^{ 3 }\\ f''(\frac { 1 }{ 3 } )\quad =\quad 720*8\\ f''(\frac { 1 }{ 3 } )\quad =\quad 5760

Luciano Canela
Jan 5, 2015

Pt: Basta aplicar a regra da cadeia duas vezes na função para encontrar a segunda derivada, e depois substituir o valor de x.

En: Basically, you can use the chain rule twice in the function and after that, you substitute the value of x in the second derivative.

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