Solve (35x^4y^9/15x^9y^12)

$$( \frac { 35 x ^ { 4 } y ^ { 9 } } { 15 x ^ { 9 } y ^ { 12 } } )$$

Evaluate

$$\frac{7}{3y^{3}x^{5}}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{35^{1}x^{4}y^{9}}{15^{1}x^{9}y^{12}}$$

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

$$\frac{35^{1}}{15^{1}}x^{4-9}y^{9-12}$$

Subtract 9 from 4.

$$\frac{35^{1}}{15^{1}}x^{-5}y^{9-12}$$

Subtract 12 from 9.

$$\frac{35^{1}}{15^{1}}\times \frac{1}{x^{5}}y^{-3}$$

Reduce the fraction \frac{35}{15} to lowest terms by extracting and canceling out 5.

$$\frac{7}{3}\times \frac{1}{x^{5}}\times \frac{1}{y^{3}}$$

Differentiate w.r.t. x

$$-\frac{35}{3y^{3}x^{6}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{35y^{9}}{15y^{12}}x^{4-9})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{7}{3y^{3}}x^{-5})$$

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}.

$$-5\times \frac{7}{3y^{3}}x^{-5-1}$$

Do the arithmetic.

$$\left(-\frac{35}{3y^{3}}\right)x^{-6}$$