Solve 25xy^2/125x^5y^7

$$\frac { 25 x y ^ { 2 } } { 125 x ^ { 5 } y ^ { 7 } }$$

Evaluate

$$\frac{1}{5x^{4}y^{5}}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{25^{1}x^{1}y^{2}}{125^{1}x^{5}y^{7}}$$

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

$$\frac{25^{1}}{125^{1}}x^{1-5}y^{2-7}$$

Subtract 5 from 1.

$$\frac{25^{1}}{125^{1}}x^{-4}y^{2-7}$$

Subtract 7 from 2.

$$\frac{25^{1}}{125^{1}}\times \frac{1}{x^{4}}y^{-5}$$

Reduce the fraction \frac{25}{125} to lowest terms by extracting and canceling out 25.

$$\frac{1}{5}\times \frac{1}{x^{4}}\times \frac{1}{y^{5}}$$

Differentiate w.r.t. x

$$-\frac{4}{5\left(xy\right)^{5}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{25y^{2}}{125y^{7}}x^{1-5})$$

Do the arithmetic.

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

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}.

$$-4\times \frac{1}{5y^{5}}x^{-4-1}$$

Do the arithmetic.

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