Solve f^prime(x)=O[x]*sqrt{x/x+2}+x*0[sqrt{x/x+2}]

$$f ^ { \prime } ( x ) = O [ x ] \cdot \sqrt { \frac { x } { x + 2 } } + x \cdot 0 [ \sqrt { \frac { x } { x + 2 } } ]$$

Solve for O (complex solution)

$$\left\{\begin{matrix}O=0\text{, }&x\neq -2\\O\in \mathrm{C}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$Ox\sqrt{\frac{x}{x+2}}+0x\sqrt{\frac{x}{x+2}}=\frac{\mathrm{d}}{\mathrm{d}x}(f)x$$

Subtract 0x\sqrt{\frac{x}{x+2}} from both sides.

$$Ox\sqrt{\frac{x}{x+2}}=\frac{\mathrm{d}}{\mathrm{d}x}(f)x-0x\sqrt{\frac{x}{x+2}}$$

Reorder the terms.

$$\sqrt{\frac{x}{x+2}}Ox=x\frac{\mathrm{d}}{\mathrm{d}x}(f)$$

The equation is in standard form.

$$\sqrt{\frac{x}{x+2}}xO=0$$

Divide 0 by x\sqrt{x\left(x+2\right)^{-1}}.

$$O=0$$

Solve for O

$$\left\{\begin{matrix}O=0\text{, }&x\geq 0\text{ or }x<-2\\O\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$Ox\sqrt{\frac{x}{x+2}}+0x\sqrt{\frac{x}{x+2}}=\frac{\mathrm{d}}{\mathrm{d}x}(f)x$$

Subtract 0x\sqrt{\frac{x}{x+2}} from both sides.

$$Ox\sqrt{\frac{x}{x+2}}=\frac{\mathrm{d}}{\mathrm{d}x}(f)x-0x\sqrt{\frac{x}{x+2}}$$

Reorder the terms.

$$\sqrt{\frac{x}{x+2}}Ox=x\frac{\mathrm{d}}{\mathrm{d}x}(f)$$

The equation is in standard form.

$$\sqrt{\frac{x}{x+2}}xO=0$$

Divide 0 by x\sqrt{x\left(x+2\right)^{-1}}.

$$O=0$$

Solve for f (complex solution)

$$f\in \mathrm{C}$$

$$\left(x\neq -2\text{ and }O=0\right)\text{ or }x=0$$

Solve for f

$$f\in \mathrm{R}$$

$$\left(x\geq 0\text{ and }O=0\right)\text{ or }\left(x<-2\text{ and }O=0\right)\text{ or }x=0$$