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Differentiate w.r.t. z

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\frac{3^{-1}\left(x^{2}\right)^{-1}y^{-1}x^{2}z}{3y^{-1}}
Expand \left(3x^{2}y\right)^{-1}.
\frac{3^{-1}x^{-2}y^{-1}x^{2}z}{3y^{-1}}
To raise a power to another power, multiply the exponents. Multiply 2 and -1 to get -2.
\frac{\frac{1}{3}x^{-2}y^{-1}x^{2}z}{3y^{-1}}
Calculate 3 to the power of -1 and get \frac{1}{3}.
\frac{\frac{1}{3}y^{-1}z}{3y^{-1}}
Multiply x^{-2} and x^{2} to get 1.
\frac{\frac{1}{3}z}{3}
Cancel out \frac{1}{y} in both numerator and denominator.
\frac{1}{9}z
Divide \frac{1}{3}z by 3 to get \frac{1}{9}z.
\frac{\mathrm{d}}{\mathrm{d}z}(\frac{3^{-1}\left(x^{2}\right)^{-1}y^{-1}x^{2}z}{3y^{-1}})
Expand \left(3x^{2}y\right)^{-1}.
\frac{\mathrm{d}}{\mathrm{d}z}(\frac{3^{-1}x^{-2}y^{-1}x^{2}z}{3y^{-1}})
To raise a power to another power, multiply the exponents. Multiply 2 and -1 to get -2.
\frac{\mathrm{d}}{\mathrm{d}z}(\frac{\frac{1}{3}x^{-2}y^{-1}x^{2}z}{3y^{-1}})
Calculate 3 to the power of -1 and get \frac{1}{3}.
\frac{\mathrm{d}}{\mathrm{d}z}(\frac{\frac{1}{3}y^{-1}z}{3y^{-1}})
Multiply x^{-2} and x^{2} to get 1.
\frac{\mathrm{d}}{\mathrm{d}z}(\frac{\frac{1}{3}z}{3})
Cancel out \frac{1}{y} in both numerator and denominator.
\frac{\mathrm{d}}{\mathrm{d}z}(\frac{1}{9}z)
Divide \frac{1}{3}z by 3 to get \frac{1}{9}z.
\frac{1}{9}z^{1-1}
The derivative of ax^{n} is nax^{n-1}.
\frac{1}{9}z^{0}
Subtract 1 from 1.
\frac{1}{9}\times 1
For any term t except 0, t^{0}=1.
\frac{1}{9}
For any term t, t\times 1=t and 1t=t.