Solve (1/2x^3)(-2/3y^2x)(-3/5y^4z)(3/10y^6{z}^2)

$$( \frac{ 1 }{ 2 } { x }^{ 3 } )(- \frac{ 2 }{ 3 } { y }^{ 2 } x)(- \frac{ 3 }{ 5 } { y }^{ 4 } z)( \frac{ 3 }{ 10 } { y }^{ 6 } { z }^{ 2 } )$$

Evaluate

$$\frac{3z^{3}x^{4}y^{12}}{50}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 3 and 1 to get 4.

$$\frac{1}{2}x^{4}\left(-\frac{2}{3}\right)y^{2}\left(-\frac{3}{5}\right)y^{4}z\times \frac{3}{10}y^{6}z^{2}$$

To multiply powers of the same base, add their exponents. Add 2 and 4 to get 6.

$$\frac{1}{2}x^{4}\left(-\frac{2}{3}\right)y^{6}\left(-\frac{3}{5}\right)z\times \frac{3}{10}y^{6}z^{2}$$

To multiply powers of the same base, add their exponents. Add 6 and 6 to get 12.

$$\frac{1}{2}x^{4}\left(-\frac{2}{3}\right)y^{12}\left(-\frac{3}{5}\right)z\times \frac{3}{10}z^{2}$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$\frac{1}{2}x^{4}\left(-\frac{2}{3}\right)y^{12}\left(-\frac{3}{5}\right)z^{3}\times \frac{3}{10}$$

Multiply \frac{1}{2} and -\frac{2}{3} to get -\frac{1}{3}.

$$-\frac{1}{3}x^{4}y^{12}\left(-\frac{3}{5}\right)z^{3}\times \frac{3}{10}$$

Multiply -\frac{1}{3} and -\frac{3}{5} to get \frac{1}{5}.

$$\frac{1}{5}x^{4}y^{12}z^{3}\times \frac{3}{10}$$

Multiply \frac{1}{5} and \frac{3}{10} to get \frac{3}{50}.

$$\frac{3}{50}x^{4}y^{12}z^{3}$$

Differentiate w.r.t. x

$$\frac{6\left(xz\right)^{3}y^{12}}{25}$$