Differentiability implies continuity, and ensures a smooth curve without sharp turns, allowing for a tangent line to exist.

There's a point $c$ where the tangent line's slope equals the secant line's slope over the interval.

Continuity ensures there are no breaks in the function, so an intermediate value must be attained.

The existence of a point $c$ in $(a,b)$ where the instantaneous rate of change equals the average rate of change over $[a,b]$.

The Mean Value Theorem is a special case of the Intermediate Value Theorem applied to the derivative of a function.

Rolle's Theorem is a special case of the Mean Value Theorem where $f(a) = f(b)$.

The function must be continuous on the closed interval $[a, b]$ and differentiable on the open interval $(a, b)$.

It guarantees that there is at least one point where the instantaneous rate of change is equal to the average rate of change over the interval.

It provides a bound on the error when approximating a function value using the average rate of change.

It can be used to relate the average velocity of an object to its instantaneous velocity at some point in time.

$f'(c)=\frac{f(b)-f(a)}{b-a}$

$\frac{f(b)-f(a)}{b-a}$

$nx^{n-1}$

Solve $f'(c) = \frac{f(b) - f(a)}{b - a}$ for $c$.

$m = \frac{f(b) - f(a)}{b - a}$

$f'(x)$

$\frac{d}{dx}(x^n) = nx^{n-1}$

$\frac{d}{dx}[cf(x)] = c \frac{d}{dx}f(x)$

$\frac{d}{dx}[f(x) \pm g(x)] = \frac{d}{dx}f(x) \pm \frac{d}{dx}g(x)$

$\frac{f(b) - f(a)}{b - a}$

If $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, there exists a $c$ in $(a,b)$ such that $f'(c)=\frac{f(b)-f(a)}{b-a}$.

No holes, asymptotes, or jump discontinuities between $a$ and $b$, including at points $a$ and $b$.

Continuous and $\lim_{x\to c} \frac{f(x) - f(c)}{x - c}$ exists for all $c$ in $(a, b)$.

The slope of the secant line between two points on a function's graph.

The slope of the tangent line at a specific point on a function's graph.

A line that intersects a curve at two or more points.

A line that touches a curve at a single point and has the same slope as the curve at that point.

An interval that includes its endpoints, denoted by square brackets [a, b].

An interval that does not include its endpoints, denoted by parentheses (a, b).

The derivative of the function exists at every point in the open interval.