Solve x^-4y^6z^-2/x^-2{y^4{z}^6}

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

Evaluate

$$\frac{\left(\frac{y}{x}\right)^{2}}{z^{8}}$$

Solution Steps

Cancel out y^{4} in both numerator and denominator.

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

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

$$\frac{y^{2}}{x^{2}z^{8}}$$

Differentiate w.r.t. x

$$-\frac{2y^{2}}{x^{3}z^{8}}$$

Solution Steps

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

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

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{y^{2}}{z^{8}}x^{-2})$$

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}.

$$-2\times \frac{y^{2}}{z^{8}}x^{-2-1}$$

Do the arithmetic.

$$\left(-\frac{2y^{2}}{z^{8}}\right)x^{-3}$$