Summary of the Derivative as a Function

The derivative of a function [latex]f(x)[/latex] is the function whose value at [latex]x[/latex] is [latex]f^{\prime}(x)[/latex].
The graph of a derivative of a function [latex]f(x)[/latex] is related to the graph of [latex]f(x)[/latex]. Where [latex]f(x)[/latex] has a tangent line with positive slope, [latex]f^{\prime}(x)>0[/latex]. Where [latex]f(x)[/latex] has a tangent line with negative slope, [latex]f^{\prime}(x)<0[/latex]. Where [latex]f(x)[/latex] has a horizontal tangent line, [latex]f^{\prime}(x)=0[/latex].
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 [latex]n\text{th}[/latex] derivative.

Key Equations

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

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 [latex]a[/latex]: a function for which [latex]f^{\prime}(a)[/latex] exists is differentiable at [latex]a[/latex]

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

differentiable function: a function for which [latex]f^{\prime}(x)[/latex] exists is a differentiable function

higher-order derivative: a derivative of a derivative, from the second derivative to the [latex]n[/latex]th derivative, is called a higher-order derivative