IVT For Derivatives

#calc/theorem
If $a$ and $b$ are any two points in an interval on which $f$ is differentiable, then $f^{\prime }$ takes on every value between $f^{\prime }(a)$ and $f^{\prime }(b)$.

Stated basically

Take two points $a$ and $b$ on a function $f$ that have nothing weird between them (non-differential spots, criteria for which can be found here). The derivative of $f$ in that range will take on all values between the derivative of $a$ and the derivative of $b$.

• This makes sense when you think about the fact that a derivative is a function too, so the Intermediate Value Theorem should apply to it too.