Solve frac{x^{-3}*y^{-4}*z^-6}{x^-2*y^-6*z^0}

$$\frac { x ^ { - 3 } \cdot y ^ { - 4 } \cdot z ^ { - 6 } } { x ^ { - 2 } \cdot y ^ { - 6 } \cdot z ^ { 0 } }$$

Evaluate

$$\frac{y^{2}}{xz^{6}}$$

Solution Steps

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

$$\frac{z^{-6}x^{-3}y^{2}}{x^{-2}z^{0}}$$

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

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

Calculate x to the power of 1 and get x.

$$\frac{y^{2}}{xz^{6}}$$

Differentiate w.r.t. x

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

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{y^{4}z^{6}\times \frac{z^{0}}{y^{6}}}x^{-3-\left(-2\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{y^{2}}{z^{6}}\times \frac{1}{x})$$

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}.

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

Do the arithmetic.

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