Summary of the Derivative as a Function

The derivative of a function $f(x)$ is the function whose value at $x$ is $f^{\prime}(x)$ .
The graph of a derivative of a function $f(x)$ is related to the graph of $f(x)$ . Where $f(x)$ has a tangent line with positive slope, $f^{\prime}(x)>0$ . Where $f(x)$ has a tangent line with negative slope, $f^{\prime}(x)<0$ . Where $f(x)$ has a horizontal tangent line, $f^{\prime}(x)=0$ .
If a function is differentiable at a point, then it is continuous at that point. A function is not differentiable at a point if it is not continuous at the point, if it has a vertical tangent line at the point, or if the graph has a sharp corner or cusp.
Higher-order derivatives are derivatives of derivatives, from the second derivative to the $n\text{th}$ derivative.

Key Equations

The derivative function
$f^{\prime}(x)=\underset{h\to 0}{\lim}\dfrac{f(x+h)-f(x)}{h}$

derivative function: gives the derivative of a function at each point in the domain of the original function for which the derivative is defined

differentiable at $a$: a function for which $f^{\prime}(a)$ exists is differentiable at $a$

differentiable on $S$: a function for which $f^{\prime}(x)$ exists for each $x$ in the open set $S$ is differentiable on $S$

differentiable function: a function for which $f^{\prime}(x)$ exists is a differentiable function

higher-order derivative: a derivative of a derivative, from the second derivative to the $n$ th derivative, is called a higher-order derivative