Solve 4x^2y^3/-6xy^5

$$\frac{ 4 { x }^{ 2 } { y }^{ 3 } }{ -6x { y }^{ 5 } }$$

Evaluate

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

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{4^{1}x^{2}y^{3}}{\left(-6\right)^{1}x^{1}y^{5}}$$

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

$$\frac{4^{1}}{\left(-6\right)^{1}}x^{2-1}y^{3-5}$$

Subtract 1 from 2.

$$\frac{4^{1}}{\left(-6\right)^{1}}x^{1}y^{3-5}$$

Subtract 5 from 3.

$$\frac{4^{1}}{\left(-6\right)^{1}}xy^{-2}$$

Reduce the fraction \frac{4}{-6} to lowest terms by extracting and canceling out 2.

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

Differentiate w.r.t. x

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

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4y^{3}}{-6y^{5}}x^{2-1})$$

Do the arithmetic.

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

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}.

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

Do the arithmetic.

$$\left(-\frac{2}{3y^{2}}\right)x^{0}$$

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

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

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

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