Solve (-2/3xyz^3)div(-3/5x^5y^4z^6)

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

Evaluate

$$\frac{10}{9x^{4}\left(yz\right)^{3}}$$

Solution Steps

Fraction \frac{-2}{3} can be rewritten as -\frac{2}{3} by extracting the negative sign.

$$\frac{-\frac{2}{3}xyz^{3}}{-\frac{3}{5}x^{5}y^{4}z^{6}}$$

Cancel out xyz^{3} in both numerator and denominator.

$$\frac{-\frac{2}{3}}{-\frac{3}{5}y^{3}z^{3}x^{4}}$$

Express \frac{-\frac{2}{3}}{-\frac{3}{5}y^{3}z^{3}x^{4}} as a single fraction.

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

Multiply 3 and -\frac{3}{5} to get -\frac{9}{5}.

$$\frac{-2}{-\frac{9}{5}y^{3}z^{3}x^{4}}$$

Differentiate w.r.t. x

$$-\frac{40}{9x^{5}\left(yz\right)^{3}}$$

Solution Steps

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

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

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{10}{9\left(yz\right)^{3}}x^{-4})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$-4\times \frac{10}{9\left(yz\right)^{3}}x^{-4-1}$$

Do the arithmetic.

$$\left(-\frac{40}{9\left(yz\right)^{3}}\right)x^{-5}$$