Solve (+16/3x^{5}y^{5}z^3):(-4/27x^3y^4z^2)

$$( + \frac { 16 } { 3 } x ^ { 5 } y ^ { 5 } z ^ { 3 } ) : ( - \frac { 4 } { 27 } x ^ { 3 } y ^ { 4 } z ^ { 2 } )$$

Evaluate

$$-36yzx^{2}$$

Solution Steps

Cancel out z^{2}x^{3}y^{4} in both numerator and denominator.

$$\frac{\frac{16}{3}yzx^{2}}{-\frac{4}{27}}$$

Divide \frac{16}{3}yzx^{2} by -\frac{4}{27} by multiplying \frac{16}{3}yzx^{2} by the reciprocal of -\frac{4}{27}.

$$\frac{\frac{16}{3}yzx^{2}\times 27}{-4}$$

Multiply \frac{16}{3} and 27 to get 144.

$$\frac{144yzx^{2}}{-4}$$

Divide 144yzx^{2} by -4 to get -36yzx^{2}.

$$-36yzx^{2}$$

Differentiate w.r.t. x

$$-72xyz$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{16z^{3}y^{5}}{3\left(-\frac{4z^{2}y^{4}}{27}\right)}x^{5-3})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(-36yz\right)x^{2})$$

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}.

$$2\left(-36yz\right)x^{2-1}$$

Do the arithmetic.

$$\left(-72yz\right)x^{1}$$

For any term t, t^{1}=t.

$$\left(-72yz\right)x$$