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Differentiate w.r.t. m
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\frac{\frac{ma^{\frac{1}{3}}}{9nb}}{\frac{\left(3n^{\frac{1}{2}}a^{\frac{3}{2}}\right)^{-2}}{\left(b^{-\frac{1}{2}}\right)^{-2}}}
To raise \frac{3n^{\frac{1}{2}}a^{\frac{3}{2}}}{b^{-\frac{1}{2}}} to a power, raise both numerator and denominator to the power and then divide.
\frac{ma^{\frac{1}{3}}\left(b^{-\frac{1}{2}}\right)^{-2}}{9nb\times \left(3n^{\frac{1}{2}}a^{\frac{3}{2}}\right)^{-2}}
Divide \frac{ma^{\frac{1}{3}}}{9nb} by \frac{\left(3n^{\frac{1}{2}}a^{\frac{3}{2}}\right)^{-2}}{\left(b^{-\frac{1}{2}}\right)^{-2}} by multiplying \frac{ma^{\frac{1}{3}}}{9nb} by the reciprocal of \frac{\left(3n^{\frac{1}{2}}a^{\frac{3}{2}}\right)^{-2}}{\left(b^{-\frac{1}{2}}\right)^{-2}}.
\frac{ma^{\frac{1}{3}}b^{1}}{9nb\times \left(3n^{\frac{1}{2}}a^{\frac{3}{2}}\right)^{-2}}
To raise a power to another power, multiply the exponents. Multiply -\frac{1}{2} and -2 to get 1.
\frac{ma^{\frac{1}{3}}b}{9nb\times \left(3n^{\frac{1}{2}}a^{\frac{3}{2}}\right)^{-2}}
Calculate b to the power of 1 and get b.
\frac{ma^{\frac{1}{3}}b}{9nb\times 3^{-2}\left(n^{\frac{1}{2}}\right)^{-2}\left(a^{\frac{3}{2}}\right)^{-2}}
Expand \left(3n^{\frac{1}{2}}a^{\frac{3}{2}}\right)^{-2}.
\frac{ma^{\frac{1}{3}}b}{9nb\times 3^{-2}n^{-1}\left(a^{\frac{3}{2}}\right)^{-2}}
To raise a power to another power, multiply the exponents. Multiply \frac{1}{2} and -2 to get -1.
\frac{ma^{\frac{1}{3}}b}{9nb\times 3^{-2}n^{-1}a^{-3}}
To raise a power to another power, multiply the exponents. Multiply \frac{3}{2} and -2 to get -3.
\frac{ma^{\frac{1}{3}}b}{9nb\times \frac{1}{9}n^{-1}a^{-3}}
Calculate 3 to the power of -2 and get \frac{1}{9}.
\frac{ma^{\frac{1}{3}}b}{nbn^{-1}a^{-3}}
Multiply 9 and \frac{1}{9} to get 1.
\frac{ma^{\frac{1}{3}}b}{ba^{-3}}
Multiply n and n^{-1} to get 1.
\frac{\sqrt[3]{a}m}{a^{-3}}
Cancel out b in both numerator and denominator.
ma^{\frac{10}{3}}
To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.