Solve (-3x^2)(2xy^2)(6y)/(8x^3y)(2xy^3)

$$\frac { ( - 3 x ^ { 2 } ) ( 2 x y ^ { 2 } ) ( 6 y ) } { ( 8 x ^ { 3 } y ) ( 2 x y ^ { 3 } ) }$$

Evaluate

$$-\frac{9}{4xy}$$

Solution Steps

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

$$\frac{-3x^{3}\times 2y^{2}\times 6y}{8x^{3}y\times 2xy^{3}}$$

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

$$\frac{-3x^{3}\times 2y^{3}\times 6}{8x^{3}y\times 2xy^{3}}$$

To multiply powers of the same base, add their exponents. Add 3 and 1 to get 4.

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

To multiply powers of the same base, add their exponents. Add 1 and 3 to get 4.

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

Cancel out 2\times 2x^{3}y^{3} in both numerator and denominator.

$$\frac{-3\times 3}{2\times 2xy}$$

Multiply -3 and 3 to get -9.

$$\frac{-9}{2\times 2xy}$$

Multiply 2 and 2 to get 4.

$$\frac{-9}{4xy}$$

Differentiate w.r.t. x

$$\frac{9}{4yx^{2}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(-\frac{36xy^{3}}{16xy^{4}}\right)x^{2-3})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(-\frac{9}{4y}\right)\times \frac{1}{x})$$

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}.

$$-\left(-\frac{9}{4y}\right)x^{-1-1}$$

Do the arithmetic.

$$\frac{9}{4y}x^{-2}$$