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To differentiate f(x)=( x ( x^4 + 2x ( 8/(x+2))^1/4 )^1/2 )^1/3 make some substitutions:

u=(8/(x+2))^¼, u=(8^¼)(x+2)^⁻¼, u'=-¼(8^¼)(x+2)^-(5/4);

v=√(x⁴+2xu), v'=½(x⁴+2xu)^⁻½(4x³+2xu'+2u);

f(x)=(xv)^⅓, f'(x)=⅓(xv)^⁻⅔(xv'+v).

To take this any further would require an April Fool!


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