What is (\partial)/(\partial x)(((3x^2)/(2x+4))^3) ?

The answer to (\partial)/(\partial x)(((3x^2)/(2x+4))^3) is (81x^5(x+4))/(2(2x+4)^2(x+2)^2)

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(\partial)/(\partial x)(((3x^2)/(2x+4))^3)

\frac{\partial}{\partial x} ((\frac{3 x ^{2}}{2 x + 4})^{3})

\frac{81 x ^{5} ( x + 4 )}{2 ( 2 x + 4 ) ^{2} ( x + 2 ) ^{2}}

Solution steps

\frac{\partial}{\partial x} ((\frac{3 x ^{2}}{2 x + 4})^{3})

Apply the chain rule:

3 (\frac{3 x ^{2}}{2 x + 4})^{2} \frac{\partial}{\partial x} (\frac{3 x ^{2}}{2 x + 4})

= 3 (\frac{3 x ^{2}}{2 x + 4})^{2} \frac{\partial}{\partial x} (\frac{3 x ^{2}}{2 x + 4})

\frac{\partial}{\partial x} (\frac{3 x ^{2}}{2 x + 4}) = \frac{3 x ( x + 4 )}{2 ( x + 2 ) ^{2}}

= 3 (\frac{3 x ^{2}}{2 x + 4})^{2} \frac{3 x ( x + 4 )}{2 ( x + 2 ) ^{2}}

Simplify

3 (\frac{3 x ^{2}}{2 x + 4})^{2} \frac{3 x ( x + 4 )}{2 ( x + 2 ) ^{2}} : \frac{81 x ^{5} ( x + 4 )}{2 ( 2 x + 4 ) ^{2} ( x + 2 ) ^{2}}

= \frac{81 x ^{5} ( x + 4 )}{2 ( 2 x + 4 ) ^{2} ( x + 2 ) ^{2}}

What is (\partial)/(\partial x)(((3x^2)/(2x+4))^3) ?
The answer to (\partial)/(\partial x)(((3x^2)/(2x+4))^3) is (81x^5(x+4))/(2(2x+4)^2(x+2)^2)