question given the graph of f(x) below, find the graph of the derivative of f(x).

Answer

The derivative $f'(x)$ represents the slope of the tangent - line to the graph of $y = f(x)$ at the point $(x,f(x))$.

For $x\lt0$, the function $f(x)$ is a horizontal line. The slope of a horizontal line is $0$. So, $f'(x)=0$ for $x\lt0$.
For $x\gt0$, the function $f(x)$ is a non - linear increasing curve. The slope of the tangent line to the curve is positive. As $x$ increases, the slope of the tangent line to the curve $y = f(x)$ is getting smaller (the curve is becoming less steep). Also, at $x = 0$, the function has a sharp turn, so the derivative does not exist at $x = 0$.

For $x\lt0$, the graph of $y = f'(x)$ is the $x$ - axis ($y = 0$).
For $x\gt0$, the graph of $y = f'(x)$ is a positive - valued curve that is decreasing and approaches $0$ as $x$ goes to infinity.

To sketch the graph of $f'(x)$:

For $x\in(-\infty,0)$, $f'(x)=0$, so the graph of $f'(x)$ lies on the $x$ - axis.
At $x = 0$, $f'(x)$ is undefined (due to the sharp corner of $f(x)$ at $x = 0$).
For $x\in(0,\infty)$, $f'(x)\gt0$ and $f'(x)$ is a decreasing function of $x$ such that $\lim_{x\rightarrow\infty}f'(x)=0$.