I have a problem that's graphed. It's linear from $(0,0)$ to $(1,1)$, then it's a horizontal line after that.
I have to find four derivatives from this, and I've never done a problem like this before. I have to find $f'(1/2)$, $\frac{d}{dx} f(e^x)$ at $x=0$, $\frac{d}{dx} f(e^{-x})$ at $x=1$, and $\frac{d}{dx} f(e^x)$ at $x=1$.
Any help is much appreciated.
This looks like an exercise in the chain rule.
First, recall that the derivative of a function is basically a fancy term for the slope at a given point. (At least, in a simplified form...) So, to find $f'(a)$, look at the slope of the graph at $x=a$. This should help with the first question.
For the rest, recall that $\frac{d}{dx}f(g(x)) = f'(g(x))\cdot g'(x)$. So, for example, $\frac{d}{dx}f(e^x) = f'(e^x)\cdot e^x$. You have the graph of the function, so you can find $f'(a)$ for a given value of $x=a$ as above. You also can evaluate $e^x$ at any point. This will help with the rest of the questions. I will warn that not all points on that function are differentiable.
If you need more guidance, let me know via a comment. I'm changing the way I leave answers to a more open-ended method, so I'm still gauging how much answering to give.
:)