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Sergeu [11.5K]
2 years ago
7

Use the four-step definition of the derivative to find f'(x) if f(x) = −4x^3 −1.

Mathematics
1 answer:
guapka [62]2 years ago
7 0

\stackrel{de finition \textit{ of a derivative as a limit}}{\lim\limits_{h\to 0}~\cfrac{f(x+h)-f(x)}{h}} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{[-4(x+h)^3-1]~~ - ~~[-4x^3-1]}{h} \\\\\\ \cfrac{[-4(x^3+3x^2h+3xh^2+h^3)-1]~~ ~~+4x^3+1}{h} \\\\\\ \cfrac{[-4x^3-12x^2h-12xh^2-4h^3-1]~~ ~~+4x^3+1}{h} \\\\\\ \cfrac{-4x^3-12x^2h-12xh^2-4h^3-1+4x^3+1}{h}\implies \cfrac{-12x^2h-12xh^2-4h^3}{h}

\cfrac{h(-12x^2-12xh-4h^2)}{h}\implies -12x^2-12xh-4h^2 \\\\\\ \lim\limits_{h\to 0}~-12x^2-12xh-4h^2\implies \lim\limits_{h\to 0}~-12x^2-12x(0)-4(0)^2 \\\\[-0.35em] ~\dotfill\\\\ ~\hfill \lim\limits_{h\to 0}~-12x^2~\hfill

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Responder:

3 / 4x + 1/2 = x

Explicación paso a paso:

Dado que:

Capacidad total = x

Volumen inicial = cierta fracción * capacidad total

Volumen adicional agregado para llenar = 1/2

Por eso,

Inicial. Volumen + volumen adicional = volumen total

De las opciones:

Un volumen inicial de 3 / 4x

Volumen adicional de 1/2

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Sumar todo hace que la ecuación:

3 / 4x + 1/2 = x

x - 3 / 4x = 1/2

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C

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3 years ago
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