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

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

Evaluate

$$\frac{z}{xy^{5}}$$

Solution Steps

Cancel out x in both numerator and denominator.

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

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

$$\frac{y^{-2}z^{1}}{xy^{3}}$$

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

$$\frac{z^{1}}{xy^{5}}$$

Calculate z to the power of 1 and get z.

$$\frac{z}{xy^{5}}$$

Differentiate w.r.t. x

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

Solution Steps

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

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

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{z}{y^{5}}\times \frac{1}{x})$$

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}.

$$-\frac{z}{y^{5}}x^{-1-1}$$

Do the arithmetic.

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