What is the second derivative of #y = lnx^2#?

Answer 1

For the first derivative, use the chain rule.

#y = ln(u)# and #u = x^2#. Then #y' = 1/u# and #u' = 2x#.
#y' = 2x xx 1/u = 2x xx 1/(x^2) = (2x)/x^2#
So, the first derivative is #(2x)/x^2#. The second derivative can be determined by differentiating the first.

By the quotient rule:

Let #y = (g(x))/(h(x))#, so that #g(x) = 2x# and #h(x) = x^2#. The derivative is given by #y' = (g'(x) xx h(x) - g(x) xx h'(x))/(h(x))^2#
The derivative of #g(x)# is #2# and the derivative of #h(x)# is #2x#.
We can now substitute into the formula and calculate. Note: the notation #y''# is used to show that we're finding the second derivative, and not the first, as would the notation #y'#. Similarly, #y'''# would signify the third derivative.
#y'' = (2 xx x^2 -2x xx 2x)/(x^2)^2#
#y'' = (2x^2 - 4x^2)/x^4#
#y'' = (-2x^2)/x^4#
#y'' = -2/x^2#
Hence, the second derivative is #y'' = -2/x^2#.

Hopefully this helps!

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Answer 2

To find the second derivative of ( y = \ln(x^2) ), first find the first derivative using the chain rule:

[ \frac{dy}{dx} = \frac{1}{x^2} \cdot 2x = \frac{2}{x} ]

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

[ \frac{d^2y}{dx^2} = -\frac{2}{x^2} ]

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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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