Fundamentals of Differentiation

The rate of change of the position at a specific time.

The integral of the velocity function.

The total change in position over a given time interval.

The average velocity of the object.

$h''(5)=120t$

$h(5)=604$

$h'(5)=620$

$\Delta t=0$

$-3\sin(\frac{\pi}{2})$

$27\sin^2(\frac{\pi}{2})\cos(\frac{\pi}{2})$

$-3\cos(\frac{\pi}{2})$

$0$

4

0

-8

12

$\frac{1}{2\sqrt{x}} - \frac{2}{x}$

$\frac{1}{2\sqrt{x}} - \frac{2}{x^2}$

$\frac{1}{2\sqrt{x}} + \frac{2}{x^2}$

$\frac{1}{2\sqrt{x}} + \frac{2}{x}$

The function is continuous and differentiable where the derivative $c'$ exists and is finite.

The function does not exist at points where the derivative $c'$ exists or is infinite.

The function is continuous and differentiable as long as $c'$ exists as a limit, less function.

The function may or may not be required to be continuous or differentiable if the derivative $c'$ is undefined.

8

$3x^2$

6

12

$-4$

$-5$

$-1$

$4$

The derivative represents the area under the curve of a function.

The derivative is always equal to the original function.

The derivative is always greater than the function itself.

The derivative can show the function's increasing or decreasing behavior.

$\frac{1}{e}$

$\frac{1}{e^2}$

$e$

$e^2$