What is (d^2)/(dx^2)((y)((3x^2+4)^{3/2}-8)) ?

The answer to (d^2)/(dx^2)((y)((3x^2+4)^{3/2}-8)) is

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(d^2)/(dx^2)((y)((3x^2+4)^{3/2}-8))

\frac{d ^{2}}{d x ^{2}} ((y) ((3 x^{2} + 4)^{\frac{3}{2}} - 8))

\frac{54 x ^{3} \frac{d y}{d x} + 54 x ^{2} y + 72 x \frac{d y}{d x} + 36 y + \frac{d ^{2} y}{d x ^{2}} ( 3 x ^{2} + 4 ) ^{2} - 8 \frac{d ^{2} y}{d x ^{2}} ( 3 x ^{2} + 4 ) ^{\frac{1}{2}}}{( 3 x ^{2} + 4 ) ^{\frac{1}{2}}}

Solution steps

\frac{d ^{2}}{d x ^{2}} ((y) ((3 x^{2} + 4)^{\frac{3}{2}} - 8))

\frac{d}{d x} (y ((3 x^{2} + 4)^{\frac{3}{2}} - 8)) = \frac{d y}{d x} ((3 x^{2} + 4)^{\frac{3}{2}} - 8) + 9 x (3 x^{2} + 4)^{\frac{1}{2}} y

= \frac{d}{d x} (\frac{d y}{d x} ((3 x^{2} + 4)^{\frac{3}{2}} - 8) + 9 x (3 x^{2} + 4)^{\frac{1}{2}} y)

\frac{d}{d x} (\frac{d y}{d x} ((3 x^{2} + 4)^{\frac{3}{2}} - 8) + 9 x (3 x^{2} + 4)^{\frac{1}{2}} y) = \frac{54 x ^{3} \frac{d y}{d x} + 54 x ^{2} y + 72 x \frac{d y}{d x} + 36 y + \frac{d ^{2} y}{d x ^{2}} ( 3 x ^{2} + 4 ) ^{2} - 8 \frac{d ^{2} y}{d x ^{2}} ( 3 x ^{2} + 4 ) ^{\frac{1}{2}}}{( 3 x ^{2} + 4 ) ^{\frac{1}{2}}}

= \frac{54 x ^{3} \frac{d y}{d x} + 54 x ^{2} y + 72 x \frac{d y}{d x} + 36 y + \frac{d ^{2} y}{d x ^{2}} ( 3 x ^{2} + 4 ) ^{2} - 8 \frac{d ^{2} y}{d x ^{2}} ( 3 x ^{2} + 4 ) ^{\frac{1}{2}}}{( 3 x ^{2} + 4 ) ^{\frac{1}{2}}}

\frac{( 1 - cos ( θ ) ) ^{2}}{2}

\frac{x + 1}{( x + 2 ) ^{3} \cdot ( x + 5 ) \cdot ( x + 6 ) ^{2}}

\frac{( x - 5 ) ^{2}}{9}

(\frac{( 4 x ^{4} - 1 ) ^{2}}{16 x ^{4}}) + 1

\frac{3 a ^{3} t}{( 7 a ^{2} + t ^{2} )}

What is (d^2)/(dx^2)((y)((3x^2+4)^{3/2}-8)) ?
The answer to (d^2)/(dx^2)((y)((3x^2+4)^{3/2}-8)) is