Solve frac{12x^{-5}y^{8}z^-6}{-4x^-3y^-4z^-2}

$$\frac { 12 x ^ { - 5 } y ^ { 8 } z ^ { - 6 } } { - 4 x ^ { - 3 } y ^ { - 4 } z ^ { - 2 } }$$

Evaluate

$$-\frac{3y^{12}}{x^{2}z^{4}}$$

Solution Steps

Cancel out 4 in both numerator and denominator.

$$\frac{3z^{-6}x^{-5}y^{8}}{-y^{-4}x^{-3}z^{-2}}$$

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

$$\frac{3z^{-6}x^{-5}y^{12}}{-x^{-3}z^{-2}}$$

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

$$\frac{3y^{12}}{-x^{2}z^{4}}$$

Cancel out -1 in both numerator and denominator.

$$\frac{-3y^{12}}{x^{2}z^{4}}$$

Differentiate w.r.t. x

$$\frac{6y^{12}}{x^{3}z^{4}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{12y^{8}}{z^{6}\left(-\frac{4}{z^{2}y^{4}}\right)}x^{-5-\left(-3\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(-\frac{3y^{12}}{z^{4}}\right)x^{-2})$$

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}.

$$-2\left(-\frac{3y^{12}}{z^{4}}\right)x^{-2-1}$$

Do the arithmetic.

$$\frac{6y^{12}}{z^{4}}x^{-3}$$