Solve frac{24x^{-5}y^{-3}z^2}{36x^-2y^3z^-2}

$$\frac { 24 x ^ { - 5 } y ^ { - 3 } z ^ { 2 } } { 36 x ^ { - 2 } y ^ { 3 } z ^ { - 2 } }$$

Evaluate

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

Solution Steps

Cancel out 12 in both numerator and denominator.

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

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

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

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

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

Differentiate w.r.t. x

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

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{24z^{2}}{y^{3}\times \frac{36y^{3}}{z^{2}}}x^{-5-\left(-2\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2z^{4}}{3y^{6}}x^{-3})$$

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}.

$$-3\times \frac{2z^{4}}{3y^{6}}x^{-3-1}$$

Do the arithmetic.

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