Solve (frac{2x^{-1}y^{2}x^-2}{6x^-1y^-4z^2})

$$( \frac { 2 x ^ { - 1 } y ^ { 2 } x ^ { - 2 } } { 6 x ^ { - 1 } y ^ { - 4 } z ^ { 2 } } )$$

Evaluate

$$\frac{y^{6}}{3\left(xz\right)^{2}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add -1 and -2 to get -3.

$$\frac{2x^{-3}y^{2}}{6x^{-1}y^{-4}z^{2}}$$

Cancel out 2 in both numerator and denominator.

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

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

$$\frac{x^{-3}y^{6}}{3\times \frac{1}{x}z^{2}}$$

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

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

Differentiate w.r.t. x

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

Solution Steps

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

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

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{y^{6}}{3\left(xz\right)^{2}}x^{0})$$

For any number a except 0, a^{0}=1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{y^{6}}{3\left(xz\right)^{2}})$$

The derivative of a constant term is 0.

$$0$$