Solve -24x^5y^2/32x^3y^6

$$\frac { - 24 x ^ { 5 } y ^ { 2 } } { 32 x ^ { 3 } y ^ { 6 } }$$

Evaluate

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

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{\left(-24\right)^{1}x^{5}y^{2}}{32^{1}x^{3}y^{6}}$$

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

$$\frac{\left(-24\right)^{1}}{32^{1}}x^{5-3}y^{2-6}$$

Subtract 3 from 5.

$$\frac{\left(-24\right)^{1}}{32^{1}}x^{2}y^{2-6}$$

Subtract 6 from 2.

$$\frac{\left(-24\right)^{1}}{32^{1}}x^{2}y^{-4}$$

Reduce the fraction \frac{-24}{32} to lowest terms by extracting and canceling out 8.

$$-\frac{3}{4}x^{2}\times \frac{1}{y^{4}}$$

Differentiate w.r.t. x

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

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(-\frac{24y^{2}}{32y^{6}}\right)x^{5-3})$$

Do the arithmetic.

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

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}.

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

Do the arithmetic.

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

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

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