Solve frac{y^{2}z^{0}*2zx^{2}y^2}{(x^2z^3)^2}

$$\frac { y ^ { 2 } z ^ { 0 } \cdot 2 z x ^ { 2 } y ^ { 2 } } { ( x ^ { 2 } z ^ { 3 } ) ^ { 2 } }$$

Evaluate

$$\frac{2y^{4}}{x^{2}z^{5}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$\frac{y^{4}z^{0}\times 2zx^{2}}{\left(x^{2}z^{3}\right)^{2}}$$

To multiply powers of the same base, add their exponents. Add 0 and 1 to get 1.

$$\frac{y^{4}z^{1}\times 2x^{2}}{\left(x^{2}z^{3}\right)^{2}}$$

Calculate z to the power of 1 and get z.

$$\frac{y^{4}z\times 2x^{2}}{\left(x^{2}z^{3}\right)^{2}}$$

Expand \left(x^{2}z^{3}\right)^{2}.

$$\frac{y^{4}z\times 2x^{2}}{\left(x^{2}\right)^{2}\left(z^{3}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{y^{4}z\times 2x^{2}}{x^{4}\left(z^{3}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$\frac{y^{4}z\times 2x^{2}}{x^{4}z^{6}}$$

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

$$\frac{2y^{4}}{x^{2}z^{5}}$$

Differentiate w.r.t. x

$$-\frac{4y^{4}}{x^{3}z^{5}}$$