# How do you find the first and second derivative of # (ln(x^(2)+3))^(3)#?

The first derivative is:

The second derivative is:

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To find the first derivative of the function ( (ln(x^{2}+3))^{3} ), apply the chain rule. The derivative of the outer function, ( u^{3} ), with respect to ( u ), is ( 3u^{2} ), where ( u = ln(x^{2}+3) ). Then, the derivative of ( ln(x^{2}+3) ) with respect to ( x ) is ( \frac{1}{x^{2}+3} ) times the derivative of ( x^{2}+3 ), which is ( 2x ).

Therefore, the first derivative of the function is ( 3(ln(x^{2}+3))^{2} * \frac{1}{x^{2}+3} * 2x ).

To find the second derivative, apply the chain rule again to differentiate ( 3(ln(x^{2}+3))^{2} ) with respect to ( x ). The derivative of ( 3(ln(x^{2}+3))^{2} ) with respect to ( u ) is ( 6u ), where ( u = ln(x^{2}+3) ). Then, differentiate ( ln(x^{2}+3) ) with respect to ( x ) to get ( \frac{1}{x^{2}+3} * 2x ) again.

Thus, the second derivative of the function is ( 6(ln(x^{2}+3)) * \frac{1}{x^{2}+3} * 2x ).

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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.

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