Solve frac{x^{2}}{14x^{3}+21x^2y^frac{0{y}}}x^3/6x^2+9xy=

$$\frac { x ^ { 2 } } { 14 x ^ { 3 } + 21 x ^ { 2 } y ^ { \frac { 0 } { y } } } \frac { x ^ { 3 } } { 6 x ^ { 2 } + 9 x y } =$$

Evaluate

$$\frac{x^{2}}{21\left(2x+3\right)\left(2x+3y\right)}$$

Short Solution Steps

Zero divided by any non-zero term gives zero.

$$\frac{x^{2}}{14x^{3}+21x^{2}y^{0}}\times \frac{x^{3}}{6x^{2}+9xy}$$

Calculate y to the power of 0 and get 1.

$$\frac{x^{2}}{14x^{3}+21x^{2}\times 1}\times \frac{x^{3}}{6x^{2}+9xy}$$

Multiply 21 and 1 to get 21.

$$\frac{x^{2}}{14x^{3}+21x^{2}}\times \frac{x^{3}}{6x^{2}+9xy}$$

Factor the expressions that are not already factored in \frac{x^{2}}{14x^{3}+21x^{2}}.

$$\frac{x^{2}}{7\left(2x+3\right)x^{2}}\times \frac{x^{3}}{6x^{2}+9xy}$$

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

$$\frac{1}{7\left(2x+3\right)}\times \frac{x^{3}}{6x^{2}+9xy}$$

Factor the expressions that are not already factored in \frac{x^{3}}{6x^{2}+9xy}.

$$\frac{1}{7\left(2x+3\right)}\times \frac{x^{3}}{3x\left(2x+3y\right)}$$

Cancel out x in both numerator and denominator.

$$\frac{1}{7\left(2x+3\right)}\times \frac{x^{2}}{3\left(2x+3y\right)}$$

Multiply \frac{1}{7\left(2x+3\right)} times \frac{x^{2}}{3\left(2x+3y\right)} by multiplying numerator times numerator and denominator times denominator.

$$\frac{x^{2}}{7\left(2x+3\right)\times 3\left(2x+3y\right)}$$

Multiply 7 and 3 to get 21.

$$\frac{x^{2}}{21\left(2x+3\right)\left(2x+3y\right)}$$

Use the distributive property to multiply 21 by 2x+3.

$$\frac{x^{2}}{\left(42x+63\right)\left(2x+3y\right)}$$

Use the distributive property to multiply 42x+63 by 2x+3y.

$$\frac{x^{2}}{84x^{2}+126xy+126x+189y}$$

Differentiate w.r.t. x

$$\frac{2x\left(xy+x+3y\right)}{7\left(\left(2x+3\right)\left(2x+3y\right)\right)^{2}}$$