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What is the second derivative of #f(x)= 3x^(2/3)-x^2#?

Answer 1

Christopher Allers

#f''(x) = -2(1/(3x^(4/3))+1)#

#f(x) = 3x^(2/3)-x^2#

#f'(x) = 3* 2/3x^(2/3-1) - 2x# [Power rule]

#= 2x^(-1/3) - 2x#

#f''(x) = 2* -1/3x^(-1/3-1) -2# [Power rule]

#= -2(1/3x^(-4/3)+1)#

#= -2(1/(3x^(4/3))+1)#

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Answer 2

Emily Ammerman

To find the second derivative of ( f(x) = 3x^{\frac{2}{3}} - x^2 ), we first need to find the first derivative and then differentiate it again.

First derivative ( f'(x) ): [ f(x) = 3x^{\frac{2}{3}} - x^2 ] [ f'(x) = \frac{2}{3} \cdot 3x^{\frac{2}{3} - 1} - 2x^{2 - 1} ] [ f'(x) = 2x^{-\frac{1}{3}} - 2x ]

Second derivative ( f''(x) ): [ f'(x) = 2x^{-\frac{1}{3}} - 2x ] [ f''(x) = -\frac{1}{3} \cdot 2x^{-\frac{1}{3} - 1} - 2 ] [ f''(x) = -\frac{2}{3}x^{-\frac{4}{3}} - 2 ] [ f''(x) = -\frac{2}{3x^{\frac{4}{3}}} - 2 ]

So, the second derivative of ( f(x) = 3x^{\frac{2}{3}} - x^2 ) is ( f''(x) = -\frac{2}{3x^{\frac{4}{3}}} - 2 ).

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Answer 3

Emilia Aguilar

The second derivative of ( f(x) = 3x^{\frac{2}{3}} - x^2 ) is ( f''(x) = -4x^{-\frac{4}{3}} - 4 ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.