Solve 3x^2y^3/3y^2x^3

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

Evaluate

$$\frac{y}{x}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{3^{1}x^{2}y^{3}}{3^{1}x^{3}y^{2}}$$

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

$$3^{1-1}x^{2-3}y^{3-2}$$

Subtract 1 from 1.

$$3^{0}x^{2-3}y^{3-2}$$

For any number a except 0, a^{0}=1.

$$x^{2-3}y^{3-2}$$

Subtract 3 from 2.

$$\frac{1}{x}y^{3-2}$$

Subtract 2 from 3.

$$\frac{1}{x}y^{1}$$

For any term t, t^{1}=t.

$$\frac{1}{x}y$$

Differentiate w.r.t. x

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

Solution Steps

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

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

Do the arithmetic.

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

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}.

$$-yx^{-1-1}$$

Do the arithmetic.

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