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

Answer 1

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

#f''(x)=((-6x^2+18)ln^2(x^2+3) + 24x ln(x^2+3))/(x^2+3)^2#

The first derivative is:

#f'(x)=3ln^2(x^2+3)*1/(x^2+3)*2x#
#=(6xln^2(x^2+3))/(x^2+3)#

The second derivative is:

#f''(x)=((6ln^2(x^2+3) + 6x*2 ln(x^2+3)* 1/(x^2+3) * 2x)*(x^2+3)-6xln^2(x^2+3)*2x)/(x^2+3)^2#
#=((6ln^2(x^2+3) + (24x ln(x^2+3))/(x^2+3))*(x^2+3)-12x^2ln^2(x^2+3))/(x^2+3)^2#
#=(((6(x^2+3)ln^2(x^2+3) + 24x ln(x^2+3))/cancel(x^2+3))*cancel((x^2+3))-12x^2ln^2(x^2+3))/(x^2+3)^2#
#=(6(x^2+3)ln^2(x^2+3) + 24x ln(x^2+3)-12x^2ln^2(x^2+3))/(x^2+3)^2#
#=((6x^2+18-12x^2)ln^2(x^2+3) + 24x ln(x^2+3))/(x^2+3)^2#
#=((-6x^2+18)ln^2(x^2+3) + 24x ln(x^2+3))/(x^2+3)^2#
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Answer 2

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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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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