What is the second derivative of #f(x)= xe^(x^3)#?
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To find the second derivative of ( f(x) = xe^{x^3} ), we first need to find the first derivative using the product rule.
( f'(x) = e^{x^3} + xe^{x^3} \cdot 3x^2 )
Now, to find the second derivative, we differentiate ( f'(x) ):
( f''(x) = (3x^2 \cdot e^{x^3}) + (xe^{x^3} \cdot 6x) + (e^{x^3} \cdot 6x) )
Simplify:
( f''(x) = 3x^2 \cdot e^{x^3} + 6x^2 \cdot e^{x^3} + 6x \cdot e^{x^3} )
( f''(x) = (3x^2 + 6x + 6x^2) \cdot e^{x^3} )
Finally, combine like terms:
( f''(x) = (9x^2 + 6x) \cdot e^{x^3} )
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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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