What is the second derivative of? : #2(x^2-1)^3#

Answer 1

#12(x^2-1)^2 + 48x^2(x^2-1)#

To find the second derivative, you need to find the first derivative, so let's do that immediately:

#d/dx 2(x^2-1)^3 = 6(x^2-1)^2 * (2x) = 12x(x^2-1)^2#

I used the chain rule and product rule to find the first derivative. Now let's find the second derivative:

#d/dx 12x(x^2-1)^2 = 12(x^2-1)^2+12x*[2(x^2-1)*2x] = 12(x^2-1)^2 + 48x^2(x^2-1)#

For the second derivative, I used the product rule and chain rule.

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Answer 2

# (d^2)/(dx^2) 2(x^2-1)^3 = 12(x^2-1)(5x^2-1) #

First we generate the first derivative using the chain rule:

# d/dx 2(x^2-1)^3 = 2(3)2(x^2-1)^2 d/dx ( x^2-1)#
# " " = 6(x^2-1)^2 (2x)#
# " " = 12x(x^2-1)^2#

Then, we generate the second derivative using the product rule and the chain rule:

# (d^2)/(dx^2) 2(x^2-1)^3 = d/dx 12x(x^2-1)^2 # # " " = 12x(d/dx (x^2-1)^2) + (d/dx12x)(x^2-1)^2 # # " " = 12x(2(x^2-1)(d/dx(x^2-1))) + 12(x^2-1)^2 # # " " = 24x(x^2-1)(2x) + 12(x^2-1)^2 #
# " " = 12(x^2-1){4x^2 + (x^2-1)} # # " " = 12(x^2-1)(5x^2-1) #
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Answer 3

To find the second derivative of (2(x^2 - 1)^3), follow these steps:

  1. Find the first derivative using the chain rule: [f'(x) = 6(x^2 - 1)^2 \cdot 2x]

  2. Simplify the first derivative: [f'(x) = 12x(x^2 - 1)^2]

  3. Now, differentiate the first derivative to find the second derivative: [f''(x) = 12(x^2 - 1)^2 + 24x^2(x^2 - 1)]

  4. Simplify the second derivative: [f''(x) = 12(x^2 - 1)^2 + 24x^2(x^2 - 1)]

Therefore, the second derivative of (2(x^2 - 1)^3) is (12(x^2 - 1)^2 + 24x^2(x^2 - 1)).

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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.

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.

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.

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.

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