Solve 4x/8x^3+12x^2ydiv12x/16x^2y^2

$$\frac { 4 x } { 8 x ^ { 3 } + 12 x ^ { 2 } y } \div \frac { 12 x } { 16 x ^ { 2 } y ^ { 2 } }$$

Evaluate

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

Short Solution Steps

Divide \frac{4x}{8x^{3}+12x^{2}y} by \frac{12x}{16x^{2}y^{2}} by multiplying \frac{4x}{8x^{3}+12x^{2}y} by the reciprocal of \frac{12x}{16x^{2}y^{2}}.

$$\frac{4x\times 16x^{2}y^{2}}{\left(8x^{3}+12x^{2}y\right)\times 12x}$$

Cancel out 4x in both numerator and denominator.

$$\frac{16x^{2}y^{2}}{3\left(8x^{3}+12yx^{2}\right)}$$

Factor the expressions that are not already factored.

$$\frac{16x^{2}y^{2}}{3\times 4\left(2x+3y\right)x^{2}}$$

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

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

Expand the expression.

$$\frac{4y^{2}}{6x+9y}$$

Differentiate w.r.t. x

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