Solve 3x^3y/9x^2y^5

$$\frac { 3 x ^ { 3 } y } { 9 x ^ { 2 } y ^ { 5 } }$$

Evaluate

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

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{3^{1}x^{3}y^{1}}{9^{1}x^{2}y^{5}}$$

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

$$\frac{3^{1}}{9^{1}}x^{3-2}y^{1-5}$$

Subtract 2 from 3.

$$\frac{3^{1}}{9^{1}}x^{1}y^{1-5}$$

Subtract 5 from 1.

$$\frac{3^{1}}{9^{1}}xy^{-4}$$

Reduce the fraction \frac{3}{9} to lowest terms by extracting and canceling out 3.

$$\frac{1}{3}x\times \frac{1}{y^{4}}$$

Differentiate w.r.t. x

$$\frac{1}{3y^{4}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3y}{9y^{5}}x^{3-2})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{3y^{4}}x^{1})$$

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}.

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

Do the arithmetic.

$$\frac{1}{3y^{4}}x^{0}$$

For any term t except 0, t^{0}=1.

$$\frac{1}{3y^{4}}\times 1$$

For any term t, t\times 1=t and 1t=t.

$$\frac{1}{3y^{4}}$$