Solve 15x^3y/45x^5y^14=

$$\frac { 15 x ^ { 3 } y } { 45 x ^ { 5 } y ^ { 14 } } =$$

Evaluate

$$\frac{1}{3x^{2}y^{13}}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{15^{1}x^{3}y^{1}}{45^{1}x^{5}y^{14}}$$

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

$$\frac{15^{1}}{45^{1}}x^{3-5}y^{1-14}$$

Subtract 5 from 3.

$$\frac{15^{1}}{45^{1}}x^{-2}y^{1-14}$$

Subtract 14 from 1.

$$\frac{15^{1}}{45^{1}}\times \frac{1}{x^{2}}y^{-13}$$

Reduce the fraction \frac{15}{45} to lowest terms by extracting and canceling out 15.

$$\frac{1}{3}\times \frac{1}{x^{2}}\times \frac{1}{y^{13}}$$

Differentiate w.r.t. x

$$-\frac{2}{3x^{3}y^{13}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{15y}{45y^{14}}x^{3-5})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{3y^{13}}x^{-2})$$

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}.

$$-2\times \frac{1}{3y^{13}}x^{-2-1}$$

Do the arithmetic.

$$\left(-\frac{2}{3y^{13}}\right)x^{-3}$$