When does the limit definition of the derivative fail to produce a finite value for $f(x)=\sqrt{x}$?

Jerk is the third derivative of displacement with respect to time. It measures the rate of change of acceleration.

Conditions: The position function $s(t)$ has a well-defined third time derivative.

Although both graphs curve upward (indicating a positive second derivative), the narrow parabola has a much more rapid increase in slope around $x=4$ compared to the wider parabola. Since the second derivative measures the rate of change of the slope, a faster change in slope results in a larger numerical value.

Conditions: Both graphs are evaluated at the same input $x=4$.; Both graphs are curving upward near $x=4$.

Write $x_2=x_1+h$ and let $h$ approach zero through nonzero values. The nearby secants illustrate the search for one limiting slope at the fixed base point.

Conditions: $x_2$ is expressed as $x_1+h$.; $h$ approaches zero through nonzero values.

The two adjacent intervals labeled $dx$ represent two equal, small steps along the input axis. They provide a geometric model for understanding why the second derivative involves differentiating the slope again with respect to $x$.

Conditions: The visualization uses enlarged steps for clarity.; Mathematically, these steps conceptually approach zero ($dx \to 0$).

The second derivative measures the instantaneous rate at which the tangent slope changes along the graph of a function. Geometrically, it tracks how fast the first derivative (the slope) is increasing or decreasing.

Conditions: The function must be twice differentiable.