Derivatives of Common Functions

Derivative:
The derivative tells us the rate of change at an exact point. It’s like finding out exactly how fast something is moving at a particular moment.
The derivative is calculated using a limit:
$f'(a) = lim_{h to 0} frac{f(a + h) – f(a)}{h}$
$“lim” means “as h gets closer to 0 on both side”$
This formula gives us the instantaneous rate of change of the function at the point $x = a$.

Constant Function:
If $f(x) = k$, where $k$ is a constant number, then the derivative is: $${}^{}$$
$f'(x) = 0$
This means there is no change in the value, like if you are standing still.

Power Function:
If $f(x) = x^n$, where $n$ is a number, then the derivative is: $${}^{}$$
$f'(x) = nx^{n1}$
Example: If $f(x) = x^2$, then $f'(x) = 2x$. This means that for any value of $x$, we can calculate how quickly $f(x)$ changes.

Square Root Function:
If $f(x) = sqrt{x}$, then the derivative is: $${}^{}$$
$f'(x) = frac{1}{2sqrt{x}}$
This formula helps us understand how quickly the value of $sqrt{x}$changes as $x$ changes.