Solve frac{48x^{-3}y^{4}z^-5}{72x^4y^-2z^0}

$$\frac { 48 x ^ { - 3 } y ^ { 4 } z ^ { - 5 } } { 72 x ^ { 4 } y ^ { - 2 } z ^ { 0 } }$$

Evaluate

$$\frac{2y^{6}}{3z^{5}x^{7}}$$

Solution Steps

Cancel out 24 in both numerator and denominator.

$$\frac{2z^{-5}x^{-3}y^{4}}{3y^{-2}z^{0}x^{4}}$$

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

$$\frac{2z^{-5}x^{-3}y^{6}}{3z^{0}x^{4}}$$

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

$$\frac{2y^{6}}{3z^{5}x^{7}}$$

Differentiate w.r.t. x

$$-\frac{14y^{6}}{3z^{5}x^{8}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{48y^{4}}{z^{5}\times \frac{72z^{0}}{y^{2}}}x^{-3-4})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2y^{6}}{3z^{5}}x^{-7})$$

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}.

$$-7\times \frac{2y^{6}}{3z^{5}}x^{-7-1}$$

Do the arithmetic.

$$\left(-\frac{14y^{6}}{3z^{5}}\right)x^{-8}$$