Solve (+25/9x^4y^4):[8/125x^4y^2]=

$$( + \frac { 25 } { 9 } x ^ { 4 } y ^ { 4 } ) : [ \frac { 8 } { 125 } x ^ { 4 } y ^ { 2 } ] =$$

Evaluate

$$\frac{3125y^{2}}{72}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{\left(\frac{25}{9}\right)^{1}x^{4}y^{4}}{\left(\frac{8}{125}\right)^{1}x^{4}y^{2}}$$

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

$$\frac{\left(\frac{25}{9}\right)^{1}}{\left(\frac{8}{125}\right)^{1}}x^{4-4}y^{4-2}$$

Subtract 4 from 4.

$$\frac{\left(\frac{25}{9}\right)^{1}}{\left(\frac{8}{125}\right)^{1}}x^{0}y^{4-2}$$

For any number a except 0, a^{0}=1.

$$\frac{\left(\frac{25}{9}\right)^{1}}{\left(\frac{8}{125}\right)^{1}}y^{4-2}$$

Subtract 2 from 4.

$$\frac{\left(\frac{25}{9}\right)^{1}}{\left(\frac{8}{125}\right)^{1}}y^{2}$$

Divide \frac{25}{9} by \frac{8}{125} by multiplying \frac{25}{9} by the reciprocal of \frac{8}{125}.

$$\frac{3125}{72}y^{2}$$

Differentiate w.r.t. y

$$\frac{3125y}{36}$$

Steps Using Definition of a Derivative

Cancel out y^{2}x^{4} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{\frac{25}{9}y^{2}}{\frac{8}{125}})$$

Divide \frac{25}{9}y^{2} by \frac{8}{125} by multiplying \frac{25}{9}y^{2} by the reciprocal of \frac{8}{125}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{\frac{25}{9}y^{2}\times 125}{8})$$

Multiply \frac{25}{9} and 125 to get \frac{3125}{9}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{\frac{3125}{9}y^{2}}{8})$$

Divide \frac{3125}{9}y^{2} by 8 to get \frac{3125}{72}y^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{3125}{72}y^{2})$$

The derivative of ax^{n} is nax^{n-1}.

$$2\times \frac{3125}{72}y^{2-1}$$

Multiply 2 times \frac{3125}{72}.

$$\frac{3125}{36}y^{2-1}$$

Subtract 1 from 2.

$$\frac{3125}{36}y^{1}$$

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

$$\frac{3125}{36}y$$