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

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

Evaluate

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

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\frac{-3x^{3}yy^{-3}}{-12x^{-3}}$$

To multiply powers of the same base, add their exponents. Add 1 and -3 to get -2.

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

Cancel out 3 in both numerator and denominator.

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

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

$$\frac{-y^{-2}x^{6}}{-4}$$

Differentiate w.r.t. x

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

Solution Steps

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

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

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x}{4y^{2}}x^{5})$$

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}.

$$5\times \frac{x}{4y^{2}}x^{5-1}$$

Do the arithmetic.

$$\frac{5x}{4y^{2}}x^{4}$$