Solve (3x^2y)^-1x^2z/3y^-1

$$\frac { ( 3 x ^ { 2 } y ) ^ { - 1 } x ^ { 2 } z } { 3 y ^ { - 1 } }$$

Evaluate

$$\frac{z}{9}$$

Solution Steps

Expand \left(3x^{2}y\right)^{-1}.

$$\frac{3^{-1}\left(x^{2}\right)^{-1}y^{-1}x^{2}z}{3y^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and -1 to get -2.

$$\frac{3^{-1}x^{-2}y^{-1}x^{2}z}{3y^{-1}}$$

Calculate 3 to the power of -1 and get \frac{1}{3}.

$$\frac{\frac{1}{3}x^{-2}y^{-1}x^{2}z}{3y^{-1}}$$

Multiply x^{-2} and x^{2} to get 1.

$$\frac{\frac{1}{3}y^{-1}z}{3y^{-1}}$$

Cancel out \frac{1}{y} in both numerator and denominator.

$$\frac{\frac{1}{3}z}{3}$$

Divide \frac{1}{3}z by 3 to get \frac{1}{9}z.

$$\frac{1}{9}z$$

Differentiate w.r.t. z

$$\frac{1}{9} \approx 0.111111111$$

Steps Using Definition of a Derivative

Expand \left(3x^{2}y\right)^{-1}.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{3^{-1}\left(x^{2}\right)^{-1}y^{-1}x^{2}z}{3y^{-1}})$$

To raise a power to another power, multiply the exponents. Multiply 2 and -1 to get -2.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{3^{-1}x^{-2}y^{-1}x^{2}z}{3y^{-1}})$$

Calculate 3 to the power of -1 and get \frac{1}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{\frac{1}{3}x^{-2}y^{-1}x^{2}z}{3y^{-1}})$$

Multiply x^{-2} and x^{2} to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{\frac{1}{3}y^{-1}z}{3y^{-1}})$$

Cancel out \frac{1}{y} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{\frac{1}{3}z}{3})$$

Divide \frac{1}{3}z by 3 to get \frac{1}{9}z.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{1}{9}z)$$

The derivative of ax^{n} is nax^{n-1}.

$$\frac{1}{9}z^{1-1}$$

Subtract 1 from 1.

$$\frac{1}{9}z^{0}$$

For any term t except 0, t^{0}=1.

$$\frac{1}{9}\times 1$$

For any term t, t\times 1=t and 1t=t.

$$\frac{1}{9}$$