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

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

Evaluate

$$\frac{y^{3}x^{8}}{z^{4}}$$

Solution Steps

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

$$\frac{\frac{1}{z}y^{3}x^{8}}{z^{3}}$$

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

$$\frac{y^{3}x^{8}}{z^{4}}$$

Differentiate w.r.t. x

$$\frac{8y^{3}x^{7}}{z^{4}}$$

Solution Steps

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

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

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{y^{3}}{z^{4}}x^{8})$$

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}.

$$8\times \frac{y^{3}}{z^{4}}x^{8-1}$$

Do the arithmetic.

$$\frac{8y^{3}}{z^{4}}x^{7}$$