Solve frac{3x^{-3}y^{2}z^-3}{27x^-1y^-4z^3}

$$\frac { 3 x ^ { - 3 } y ^ { 2 } z ^ { - 3 } } { 27 x ^ { - 1 } y ^ { - 4 } z ^ { 3 } }$$

Evaluate

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

Solution Steps

Cancel out 3 in both numerator and denominator.

$$\frac{x^{-3}z^{-3}y^{2}}{9y^{-4}\times \frac{1}{x}z^{3}}$$

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

$$\frac{x^{-3}z^{-3}y^{6}}{9\times \frac{1}{x}z^{3}}$$

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

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

Differentiate w.r.t. x

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

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3y^{2}}{z^{3}\times \frac{27z^{3}}{y^{4}}}x^{-3-\left(-1\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{y^{6}}{9z^{6}}x^{-2})$$

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}.

$$-2\times \frac{y^{6}}{9z^{6}}x^{-2-1}$$

Do the arithmetic.

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