Solve 8xydiv(2x/3+2x/5y)

$$8 x y \div ( \frac { 2 x } { 3 } + \frac { 2 x } { 5 y } )$$

Evaluate

$$\frac{60y^{2}}{5y+3}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 5y is 15y. Multiply \frac{2x}{3} times \frac{5y}{5y}. Multiply \frac{2x}{5y} times \frac{3}{3}.

$$\frac{8xy}{\frac{2x\times 5y}{15y}+\frac{3\times 2x}{15y}}$$

Since \frac{2x\times 5y}{15y} and \frac{3\times 2x}{15y} have the same denominator, add them by adding their numerators.

$$\frac{8xy}{\frac{2x\times 5y+3\times 2x}{15y}}$$

Do the multiplications in 2x\times 5y+3\times 2x.

$$\frac{8xy}{\frac{10xy+6x}{15y}}$$

Divide 8xy by \frac{10xy+6x}{15y} by multiplying 8xy by the reciprocal of \frac{10xy+6x}{15y}.

$$\frac{8xy\times 15y}{10xy+6x}$$

Factor the expressions that are not already factored.

$$\frac{8\times 15xy^{2}}{2x\left(5y+3\right)}$$

Cancel out 2x in both numerator and denominator.

$$\frac{4\times 15y^{2}}{5y+3}$$

Expand the expression.

$$\frac{60y^{2}}{5y+3}$$

Differentiate w.r.t. y

$$\frac{60y\left(5y+6\right)}{\left(5y+3\right)^{2}}$$

Steps Using Derivative Rule for Quotient

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 5y is 15y. Multiply \frac{2x}{3} times \frac{5y}{5y}. Multiply \frac{2x}{5y} times \frac{3}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{8xy}{\frac{2x\times 5y}{15y}+\frac{3\times 2x}{15y}})$$

Since \frac{2x\times 5y}{15y} and \frac{3\times 2x}{15y} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{8xy}{\frac{2x\times 5y+3\times 2x}{15y}})$$

Do the multiplications in 2x\times 5y+3\times 2x.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{8xy}{\frac{10xy+6x}{15y}})$$

Divide 8xy by \frac{10xy+6x}{15y} by multiplying 8xy by the reciprocal of \frac{10xy+6x}{15y}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{8xy\times 15y}{10xy+6x})$$

Factor the expressions that are not already factored in \frac{8xy\times 15y}{10xy+6x}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{8\times 15xy^{2}}{2x\left(5y+3\right)})$$

Cancel out 2x in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{4\times 15y^{2}}{5y+3})$$

Multiply 4 and 15 to get 60.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{60y^{2}}{5y+3})$$

For any two differentiable functions, the derivative of the quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the denominator squared.

$$\frac{\left(5y^{1}+3\right)\frac{\mathrm{d}}{\mathrm{d}y}(60y^{2})-60y^{2}\frac{\mathrm{d}}{\mathrm{d}y}(5y^{1}+3)}{\left(5y^{1}+3\right)^{2}}$$

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}.

$$\frac{\left(5y^{1}+3\right)\times 2\times 60y^{2-1}-60y^{2}\times 5y^{1-1}}{\left(5y^{1}+3\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(5y^{1}+3\right)\times 120y^{1}-60y^{2}\times 5y^{0}}{\left(5y^{1}+3\right)^{2}}$$

Expand using distributive property.

$$\frac{5y^{1}\times 120y^{1}+3\times 120y^{1}-60y^{2}\times 5y^{0}}{\left(5y^{1}+3\right)^{2}}$$

To multiply powers of the same base, add their exponents.

$$\frac{5\times 120y^{1+1}+3\times 120y^{1}-60\times 5y^{2}}{\left(5y^{1}+3\right)^{2}}$$

Do the arithmetic.

$$\frac{600y^{2}+360y^{1}-300y^{2}}{\left(5y^{1}+3\right)^{2}}$$

Combine like terms.

$$\frac{\left(600-300\right)y^{2}+360y^{1}}{\left(5y^{1}+3\right)^{2}}$$

Subtract 300 from 600.

$$\frac{300y^{2}+360y^{1}}{\left(5y^{1}+3\right)^{2}}$$

Factor out 60y.

$$\frac{60y\left(5y^{1}+6y^{0}\right)}{\left(5y^{1}+3\right)^{2}}$$

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

$$\frac{60y\left(5y+6y^{0}\right)}{\left(5y+3\right)^{2}}$$

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

$$\frac{60y\left(5y+6\times 1\right)}{\left(5y+3\right)^{2}}$$

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

$$\frac{60y\left(5y+6\right)}{\left(5y+3\right)^{2}}$$