# Differentiability

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- Understand what "differentiable" means and how it is similar and different from "continuous"
- See visual cues from graphs that help us define and understand differentiability
- Define differentiability using a limit definition

Not all functions have well-defined slope functions. This lesson explores the relationship between whether a function has a well-defined slope function and the characteristics of that function.

Practice problems and worksheet coming soon!

## Smooth AF

While continuity describes a function's "connectedness", differentiability describes a function's "smoothness". Of course, like continuity, we should know the technical definition, since we'll be expected to know and use it.## Looking for Smooth

A graph is an incredibly good tool to help us identify locations of discontinuity. First and foremost, any place where a discontinuity exists will also be a place that is not differentiable. Second, places that are continuous but not differentiable tend to have a certain look to them - they are often corners, cusps, or peaks.Take a look at the following function graph.Since this graph has a jump discontinuity at $x=3$, we know the function is already not continuous at that $x$ value, and therefore it is also not differentiable there either.Other less obvious graphical indicators of places where a function is not differentiable include cusps and corners. Cusps tend to look like a 1st grader's drawing of a seagull:Each side of the cusp has a tangent slope that approaches $\infty$ or $-\infty$.Corners are like cusps but don't need to have vertical tangents. A classic example is the graph of the function $f(x) = |x|$:While the graph does not seem to have any vertical tangency behavior, there is clearly a corner where the graph does not exhibit smoothness, namely at $x=0$.## What's the "Point"?

For this topic, we will be expected to be able to do a few things - first, conceptually understand differentiability and be able to explain what it is in either words or math symbols. Next, we should be ready to show or demonstrate differentiability using either a graph or algebra approach. Finally, we should understand the concept of differentiability as a yes / no question. It is a "yes" at the place when the function is smooth, and a "no" at the places where the function has discontinuities, corners, or cusps.The common questions for quizzes and tests involve demonstrating whether a function is differentiable by applying either the graph approach or the algebra approach.The Graph ApproachThe graph approach is a strong tool but it's not perfect. If a graph is not provided and it seems sufficiently annoying or difficult to make your own graph, don't bother. Additionally, there can be cases where the graph does not seem to indicate an issue even though one exists. For example, examine the graph of $f(x) = x^{3/5}$:At $x=0$, there is a vertical tangent line, and vertical tangent lines are places that are smooth-looking yet not differentiable. This is one reason that, while graphs are extremely helpful for identifying differentiability issues, we must always be able to rely on the mathematical limit definition.- Differentiability refers to smoothness, which happens when $f'(x)$ is continuous.
- To find differentiability issues for a given function, find $f'(x)$ and look for domain issues.

## Typical Functions

As with continuity and other core derivative skills, the core function families have common and well-behaved properties that we should know and understand.- Polynomial
- Trigonometric
- Exponential
- Logarithmic
- Rational

## Mr. Math Makes It Mean

There are a few ways teachers can ask about differentiability in less-than-direct ways.Combinations of FunctionsFor any type of function combination, whether it's arithmetic of functions or composition of functions, the best plan is to consider the final product and not focus on the individual functions that are being used. Sometimes it will be enough to inspect the combination function, but we always have the option of explicitly taking its derivative and looking for continuity issues.The only thing about the foundational functions that we need to pay attention to is the domain restrictions, because we can't eliminate domain problems via composition. This next example will remind us of that.## Unknown Coefficients Problems

Teachers love to test your knowledge of differentiability with piecewise functions. One common way they do this is to ask you to find the unknown coefficients in a piecewise function such that the function is differentiable at the endpoints of the piecewise interval.## Put It To The Test

Make sure you can do the various types of problems that teachers like to ask for on quizzes and tests!- Understand differentiability as both an intuitive measure of smoothness and as a limit definition
- Know how to use graphs to evaluate differentiability and know the limitations of using a graph based approach
- Be able to evaluate where a function is and is not differentiable using the Calculus approach without graphs.
- Know the differentiation properties of common function families
- Approach combinations of functions properly, examining the final result which remembering domain restrictions on either standalone function
- Be ready to answer questions about piecewise functions, including missing coefficient problems

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