Solve (frac{x^{-2}y^5z^-3}{x^-7y^-4z})

$$( \frac { x ^ { - 2 } y ^ { 5 } z ^ { - 3 } } { x ^ { - 7 } y ^ { - 4 } z } )$$

Evaluate

$$\frac{x^{5}y^{9}}{z^{4}}$$

Solution Steps

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

$$\frac{z^{-3}x^{5}y^{9}}{z}$$

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

$$\frac{x^{5}y^{9}}{z^{4}}$$

Differentiate w.r.t. x

$$5\times \left(\frac{x}{z}\right)^{4}y^{9}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{y^{5}}{z^{3}\times \frac{z}{y^{4}}}x^{-2-\left(-7\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{y^{9}}{z^{4}}x^{5})$$

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}.

$$5\times \frac{y^{9}}{z^{4}}x^{5-1}$$

Do the arithmetic.

$$\frac{5y^{9}}{z^{4}}x^{4}$$