Find from the definition, the derivative of x^2 2x .

To find the derivative of the function $f(x)=x^2-2x$ from the definition, we use the first principle of differentiation, which is given by:

$$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$$

Step 1: Write $f(x+h)$

Substitute $x+h$ into the function:

$$f(x+h)=(x+h{)}^2-2(x+h)$$

Expanding this:

$$f(x+h)=x^2+2xh+h^2-2x-2h$$

Step 2: Compute $f(x+h)-f(x)$

Subtract $f(x)=x^2-2x$ from $f(x+h)$:

$$f(x+h)-f(x)=(x^2+2xh+h^2-2x-2h)-(x^2-2x)$$

Simplify:

$$f(x+h)-f(x)=2xh+h^2-2h$$

Step 3: Divide by $h$

$$\frac{f(x+h)-f(x)}{h}=\frac{2xh+h^2-2h}{h}=2x+h-2$$

Step 4: Take the limit as $h\to 0$

$$f^{\prime }(x)=\underset{h\to 0}{\lim }(2x+h-2)=2x-2$$

Final Answer

The derivative of $f(x)=x^2-2x$ is:

$$\boxed{2x-2}$$