Solve (-2x^-3y)(-12x^-4y^-2)/6xy^-3

$$\frac { ( - 2 x ^ { - 3 } y ) ( - 12 x ^ { - 4 } y ^ { - 2 } ) } { 6 x y ^ { - 3 } }$$

Evaluate

$$\frac{4y^{2}}{x^{8}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add -3 and -4 to get -7.

$$\frac{-2x^{-7}y\left(-12\right)y^{-2}}{6xy^{-3}}$$

To multiply powers of the same base, add their exponents. Add 1 and -2 to get -1.

$$\frac{-2x^{-7}y^{-1}\left(-12\right)}{6xy^{-3}}$$

Cancel out 2\times 3 in both numerator and denominator.

$$\frac{-4\left(-1\right)x^{-7}\times \frac{1}{y}}{y^{-3}x}$$

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

$$\frac{-4\left(-1\right)x^{-7}y^{2}}{x}$$

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

$$\frac{-4\left(-1\right)y^{2}}{x^{8}}$$

Multiply -4 and -1 to get 4.

$$\frac{4y^{2}}{x^{8}}$$

Differentiate w.r.t. x

$$-\frac{32y^{2}}{x^{9}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(-\left(-\frac{24y}{y^{2}x^{4}}\right)\right)y^{3}}{6}x^{-3-1})$$

Do the arithmetic.

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

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}.

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

Do the arithmetic.

$$\left(-\frac{16y^{2}}{x^{4}}\right)x^{-5}$$