Solve 5xy^6/15x^10y^6

$$\frac { 5 x y ^ { 6 } } { 15 x ^ { 10 } y ^ { 6 } }$$

Evaluate

$$\frac{1}{3x^{9}}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{5^{1}x^{1}y^{6}}{15^{1}x^{10}y^{6}}$$

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

$$\frac{5^{1}}{15^{1}}x^{1-10}y^{6-6}$$

Subtract 10 from 1.

$$\frac{5^{1}}{15^{1}}x^{-9}y^{6-6}$$

Subtract 6 from 6.

$$\frac{5^{1}}{15^{1}}\times \frac{1}{x^{9}}y^{0}$$

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

$$\frac{5^{1}}{15^{1}}\times \frac{1}{x^{9}}$$

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

$$\frac{1}{3}\times \frac{1}{x^{9}}$$

Differentiate w.r.t. x

$$-\frac{3}{x^{10}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5y^{6}}{15y^{6}}x^{1-10})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{3}x^{-9})$$

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}.

$$-9\times \frac{1}{3}x^{-9-1}$$

Do the arithmetic.

$$-3x^{-10}$$