Question

determine whether the function is differentiable (curve has a tangent line) at the indicated point. If it does, find its derivative. if not explain why not.

Answer 1

Let's see the sketch:

The function is differentiable at x = 0 as you can see from the graph. It doesn't have any cusp or discontinuity.

Now, to find the derivate, we use the bottom function (as x = 0 falls in this).

So,

$\begin{gathered} f(x)=x^2-x \\ f^{\prime}(x)=2x-1 \\ f^{\prime}(0)=2(0)\text{ -1} \\ f^{\prime}(0)=-1 \end{gathered}$

The derivative is -1