Solve 54x^3y^3/12x^4y

$$\frac { 54 x ^ { 3 } y ^ { 3 } } { 12 x ^ { 4 } y }$$

Evaluate

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

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{54^{1}x^{3}y^{3}}{12^{1}x^{4}y^{1}}$$

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

$$\frac{54^{1}}{12^{1}}x^{3-4}y^{3-1}$$

Subtract 4 from 3.

$$\frac{54^{1}}{12^{1}}\times \frac{1}{x}y^{3-1}$$

Subtract 1 from 3.

$$\frac{54^{1}}{12^{1}}\times \frac{1}{x}y^{2}$$

Reduce the fraction \frac{54}{12} to lowest terms by extracting and canceling out 6.

$$\frac{9}{2}\times \frac{1}{x}y^{2}$$

Differentiate w.r.t. x

$$-\frac{9\times \left(\frac{y}{x}\right)^{2}}{2}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{54y^{3}}{12y}x^{3-4})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{9y^{2}}{2}\times \frac{1}{x})$$

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}.

$$-\frac{9y^{2}}{2}x^{-1-1}$$

Do the arithmetic.

$$\left(-\frac{9y^{2}}{2}\right)x^{-2}$$