Solve 32x^5y^7/-4x^2y^9

$$\frac { 32 x ^ { 5 } y ^ { 7 } } { - 4 x ^ { 2 } y ^ { 9 } }$$

Evaluate

$$-\frac{8x^{3}}{y^{2}}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{32^{1}x^{5}y^{7}}{\left(-4\right)^{1}x^{2}y^{9}}$$

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

$$\frac{32^{1}}{\left(-4\right)^{1}}x^{5-2}y^{7-9}$$

Subtract 2 from 5.

$$\frac{32^{1}}{\left(-4\right)^{1}}x^{3}y^{7-9}$$

Subtract 9 from 7.

$$\frac{32^{1}}{\left(-4\right)^{1}}x^{3}y^{-2}$$

Divide 32 by -4.

$$-8x^{3}\times \frac{1}{y^{2}}$$

Differentiate w.r.t. x

$$-24\times \left(\frac{x}{y}\right)^{2}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{32y^{7}}{-4y^{9}}x^{5-2})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(-\frac{8}{y^{2}}\right)x^{3})$$

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}.

$$3\left(-\frac{8}{y^{2}}\right)x^{3-1}$$

Do the arithmetic.

$$\left(-\frac{24}{y^{2}}\right)x^{2}$$