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

Answer 1

Isabella Ackerman

Differentiate once, and then differentiate again.

By the quotient rule:

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

#f'(x) = (2x^2 + 6x - x^2)//(x + 3)^2#

#f'(x) = (x^2 + 6x)/(x^2 + 6x + 9)#

Differentiate once more.

#f''(x) = ((2x + 6)(x^2+ 6x + 9) - ((2x + 6)(x^2 + 6x)))/(x^2 + 6x + 9)^2#

You can simplify this further, but I'll leave the algebra up to you.

Hopefully this helps!

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Answer 2

Ezekiel Alexander

To find the second derivative of ( f(x) = \frac{x^2}{x+3} ), we first find the first derivative using the quotient rule:

[ f'(x) = \frac{(x+3)(2x) - (x^2)(1)}{(x+3)^2} ]

Simplify the expression:

[ f'(x) = \frac{2x^2 + 6x - x^2}{(x+3)^2} = \frac{x^2 + 6x}{(x+3)^2} ]

Now, differentiate ( f'(x) ) with respect to ( x ) to find the second derivative:

[ f''(x) = \frac{(x+3)^2(2x) - (x^2 + 6x)(2(x+3))}{(x+3)^4} ]

Simplify the expression:

[ f''(x) = \frac{2x(x^2 + 6x) - 2(x^2 + 6x)(x+3)}{(x+3)^3} ]

[ f''(x) = \frac{2x^3 + 12x^2 - 2x^3 - 12x^2 - 12x}{(x+3)^3} ]

[ f''(x) = \frac{-12x}{(x+3)^3} ]

So, the second derivative of ( f(x) = \frac{x^2}{x+3} ) is ( f''(x) = \frac{-12x}{(x+3)^3} ).

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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.