Solve (x^-3y^-1z^-4/x^-5{y^-1{z}^-3})

$$( \frac{ { x }^{ -3 } { y }^{ -1 } { z }^{ -4 } }{ { x }^{ -5 } { y }^{ -1 } { z }^{ -3 } } )$$

Evaluate

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

Solution Steps

Cancel out \frac{1}{y} in both numerator and denominator.

$$\frac{z^{-4}x^{-3}}{x^{-5}z^{-3}}$$

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

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

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

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

Calculate z to the power of 1 and get z.

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

Differentiate w.r.t. x

$$\frac{2x}{z}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{yz^{4}\times \frac{1}{yz^{3}}}x^{-3-\left(-5\right)})$$

Do the arithmetic.

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

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}.

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

Do the arithmetic.

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

For any term t, t^{1}=t.

$$\frac{2}{z}x$$