Solve (-9/4x^5y^3z^2)div(21/2x^2{y}^2{z}^2)

$$(- \frac{ 9 }{ 4 } { x }^{ 5 } { y }^{ 3 } { z }^{ 2 } ) \div ( \frac{ 21 }{ 2 } { x }^{ 2 } { y }^{ 2 } { z }^{ 2 } )$$

Evaluate

$$-\frac{3yx^{3}}{14}$$

Solution Steps

Cancel out x^{2}y^{2}z^{2} in both numerator and denominator.

$$\frac{-\frac{9}{4}yx^{3}}{\frac{21}{2}}$$

Divide -\frac{9}{4}yx^{3} by \frac{21}{2} by multiplying -\frac{9}{4}yx^{3} by the reciprocal of \frac{21}{2}.

$$\frac{-\frac{9}{4}yx^{3}\times 2}{21}$$

Multiply -\frac{9}{4} and 2 to get -\frac{9}{2}.

$$\frac{-\frac{9}{2}yx^{3}}{21}$$

Divide -\frac{9}{2}yx^{3} by 21 to get -\frac{3}{14}yx^{3}.

$$-\frac{3}{14}yx^{3}$$

Differentiate w.r.t. x

$$-\frac{9yx^{2}}{14}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2\left(-\frac{9z^{2}y^{3}}{4}\right)}{21y^{2}z^{2}}x^{5-2})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(-\frac{3y}{14}\right)x^{3})$$

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}.

$$3\left(-\frac{3y}{14}\right)x^{3-1}$$

Do the arithmetic.

$$\left(-\frac{9y}{14}\right)x^{2}$$