Solve 18x^5y/-54x^3y^7

$$\frac { 18 x ^ { 5 } y } { - 54 x ^ { 3 } y ^ { 7 } }$$

Evaluate

$$-\frac{x^{2}}{3y^{6}}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{18^{1}x^{5}y^{1}}{\left(-54\right)^{1}x^{3}y^{7}}$$

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

$$\frac{18^{1}}{\left(-54\right)^{1}}x^{5-3}y^{1-7}$$

Subtract 3 from 5.

$$\frac{18^{1}}{\left(-54\right)^{1}}x^{2}y^{1-7}$$

Subtract 7 from 1.

$$\frac{18^{1}}{\left(-54\right)^{1}}x^{2}y^{-6}$$

Reduce the fraction \frac{18}{-54} to lowest terms by extracting and canceling out 18.

$$-\frac{1}{3}x^{2}\times \frac{1}{y^{6}}$$

Differentiate w.r.t. x

$$-\frac{2x}{3y^{6}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{18y}{-54y^{7}}x^{5-3})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(-\frac{1}{3y^{6}}\right)x^{2})$$

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}.

$$2\left(-\frac{1}{3y^{6}}\right)x^{2-1}$$

Do the arithmetic.

$$\left(-\frac{2}{3y^{6}}\right)x^{1}$$

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

$$\left(-\frac{2}{3y^{6}}\right)x$$