The second order derivative of the function \( y=\log(4x^{3}+6) \) with respect to 'x' will be:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( \displaystyle\frac{12x(3-x^{3})}{(2x^{3}+3)^{2}} \)
By the chain rule, and cancelling the common factor 2:
\( \displaystyle \frac{dy}{dx}=\frac{12x^{2}}{4x^{3}+6} \)
\( \displaystyle =\frac{6x^{2}}{2x^{3}+3} \)
Now the quotient rule:
\( \displaystyle \frac{d^{2}y}{dx^{2}}=\frac{(2x^{3}+3)\cdot 12x-6x^{2}\cdot 6x^{2}}{(2x^{3}+3)^{2}} \)
\( \displaystyle =\frac{24x^{4}+36x-36x^{4}}{(2x^{3}+3)^{2}} \)
\( \displaystyle =\frac{36x-12x^{4}}{(2x^{3}+3)^{2}} \)
\( \displaystyle =\frac{12x(3-x^{3})}{(2x^{3}+3)^{2}} \)