Derivative of $$$\frac{x^{2}}{x + 10}$$$

Apply the quotient rule $$$\frac{d}{dx} \left(\frac{f{\left(x \right)}}{g{\left(x \right)}}\right) = \frac{\frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} - f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right)}{g^{2}{\left(x \right)}}$$$ with $$$f{\left(x \right)} = x^{2}$$$ and $$$g{\left(x \right)} = x + 10$$$:

$${\color{red}\left(\frac{d}{dx} \left(\frac{x^{2}}{x + 10}\right)\right)} = {\color{red}\left(\frac{\frac{d}{dx} \left(x^{2}\right) \left(x + 10\right) - x^{2} \frac{d}{dx} \left(x + 10\right)}{\left(x + 10\right)^{2}}\right)}$$

The derivative of a sum/difference is the sum/difference of derivatives:

$$\frac{- x^{2} {\color{red}\left(\frac{d}{dx} \left(x + 10\right)\right)} + \left(x + 10\right) \frac{d}{dx} \left(x^{2}\right)}{\left(x + 10\right)^{2}} = \frac{- x^{2} {\color{red}\left(\frac{d}{dx} \left(x\right) + \frac{d}{dx} \left(10\right)\right)} + \left(x + 10\right) \frac{d}{dx} \left(x^{2}\right)}{\left(x + 10\right)^{2}}$$

The derivative of a constant is $$$0$$$:

$$\frac{- x^{2} \left({\color{red}\left(\frac{d}{dx} \left(10\right)\right)} + \frac{d}{dx} \left(x\right)\right) + \left(x + 10\right) \frac{d}{dx} \left(x^{2}\right)}{\left(x + 10\right)^{2}} = \frac{- x^{2} \left({\color{red}\left(0\right)} + \frac{d}{dx} \left(x\right)\right) + \left(x + 10\right) \frac{d}{dx} \left(x^{2}\right)}{\left(x + 10\right)^{2}}$$

Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 2$$$:

$$\frac{- x^{2} \frac{d}{dx} \left(x\right) + \left(x + 10\right) {\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)}}{\left(x + 10\right)^{2}} = \frac{- x^{2} \frac{d}{dx} \left(x\right) + \left(x + 10\right) {\color{red}\left(2 x\right)}}{\left(x + 10\right)^{2}}$$

Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 1$$$, in other words, $$$\frac{d}{dx} \left(x\right) = 1$$$:

$$\frac{- x^{2} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} + 2 x \left(x + 10\right)}{\left(x + 10\right)^{2}} = \frac{- x^{2} {\color{red}\left(1\right)} + 2 x \left(x + 10\right)}{\left(x + 10\right)^{2}}$$

Simplify:

$$\frac{- x^{2} + 2 x \left(x + 10\right)}{\left(x + 10\right)^{2}} = \frac{x \left(x + 20\right)}{\left(x + 10\right)^{2}}$$

Thus, $$$\frac{d}{dx} \left(\frac{x^{2}}{x + 10}\right) = \frac{x \left(x + 20\right)}{\left(x + 10\right)^{2}}$$$.