Solve 30x^4y^5/20x^3y^2

$$\frac { 30 x ^ { 4 } y ^ { 5 } } { 20 x ^ { 3 } y ^ { 2 } }$$

Evaluate

$$\frac{3xy^{3}}{2}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{30^{1}x^{4}y^{5}}{20^{1}x^{3}y^{2}}$$

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

$$\frac{30^{1}}{20^{1}}x^{4-3}y^{5-2}$$

Subtract 3 from 4.

$$\frac{30^{1}}{20^{1}}x^{1}y^{5-2}$$

Subtract 2 from 5.

$$\frac{30^{1}}{20^{1}}xy^{3}$$

Reduce the fraction \frac{30}{20} to lowest terms by extracting and canceling out 10.

$$\frac{3}{2}xy^{3}$$

Differentiate w.r.t. x

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

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{30y^{5}}{20y^{2}}x^{4-3})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3y^{3}}{2}x^{1})$$

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}.

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

Do the arithmetic.

$$\frac{3y^{3}}{2}x^{0}$$

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

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

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

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