Solve x^2/x+2/frac{x^3{2x+4}}

$$\frac { \frac { x ^ { 2 } } { x + 2 } } { \frac { x ^ { 3 } } { 2 x + 4 } }$$

Evaluate

$$\frac{2}{x}$$

Short Solution Steps

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

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

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

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

Factor the expressions that are not already factored.

$$\frac{2\left(x+2\right)}{x\left(x+2\right)}$$

Cancel out x+2 in both numerator and denominator.

$$\frac{2}{x}$$

Differentiate w.r.t. x

$$-\frac{2}{x^{2}}$$

Steps Using Definition of a Derivative

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

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

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x+4}{x\left(x+2\right)})$$

Factor the expressions that are not already factored in \frac{2x+4}{x\left(x+2\right)}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2\left(x+2\right)}{x\left(x+2\right)})$$

Cancel out x+2 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2}{x})$$

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

$$-2x^{-1-1}$$

Subtract 1 from -1.

$$-2x^{-2}$$