Solve (frac{Ax^{2}y^5z}{y2x^4yz^-3})^-1

$$( \frac { A x ^ { 2 } y ^ { 5 } z } { y 2 x ^ { 4 } y z ^ { - 3 } } ) ^ { - 1 }$$

Evaluate

$$\frac{y_{2}x^{2}}{A\left(yz\right)^{4}}$$

Solution Steps

Cancel out yx^{2} in both numerator and denominator.

$$\left(\frac{Azy^{4}}{z^{-3}y_{2}x^{2}}\right)^{-1}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\left(\frac{Ay^{4}z^{4}}{y_{2}x^{2}}\right)^{-1}$$

To raise \frac{Ay^{4}z^{4}}{y_{2}x^{2}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(Ay^{4}z^{4}\right)^{-1}}{\left(y_{2}x^{2}\right)^{-1}}$$

Expand \left(Ay^{4}z^{4}\right)^{-1}.

$$\frac{A^{-1}\left(y^{4}\right)^{-1}\left(z^{4}\right)^{-1}}{\left(y_{2}x^{2}\right)^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and -1 to get -4.

$$\frac{A^{-1}y^{-4}\left(z^{4}\right)^{-1}}{\left(y_{2}x^{2}\right)^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and -1 to get -4.

$$\frac{A^{-1}y^{-4}z^{-4}}{\left(y_{2}x^{2}\right)^{-1}}$$

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

$$\frac{A^{-1}y^{-4}z^{-4}}{y_{2}^{-1}\left(x^{2}\right)^{-1}}$$

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

$$\frac{A^{-1}y^{-4}z^{-4}}{y_{2}^{-1}x^{-2}}$$

Differentiate w.r.t. x

$$\frac{2xy_{2}}{A\left(yz\right)^{4}}$$