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

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

Evaluate

$$\frac{10z^{2}}{9y^{3}x^{5}}$$

Solution Steps

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

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

Cancel out xyz in both numerator and denominator.

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

Differentiate w.r.t. x

$$-\frac{50z^{2}}{9y^{3}x^{6}}$$

Solution Steps

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

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

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{10z^{2}}{9y^{3}}x^{-5})$$

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}.

$$-5\times \frac{10z^{2}}{9y^{3}}x^{-5-1}$$

Do the arithmetic.

$$\left(-\frac{50z^{2}}{9y^{3}}\right)x^{-6}$$