Differentiable Iff Partial Derivatives Exist and Are Continuous
Have you ever wondered what makes a function differentiable?
Jenn, Founder Calcworkshop®, 15+ Years Experience (Licensed & Certified Teacher)
A function is formally considered differentiable if its derivative exists at each point in its domain, but what does this mean?
It means that a function is differentiable everywhere its derivative is defined.
So, as long as you can evaluate the derivative at every point on the curve, the function is differentiable.
How To Determine Differentiability
By using limits and continuity!
The definition of differentiability is expressed as follows:
- f is differentiable on an open interval (a,b) if \(\lim _{h \rightarrow 0} \frac{f(c+h)-f(c)}{h}\) exists for every c in (a,b).
- f is differentiable, meaning \(f^{\prime}(c)\) exists, then f is continuous at c.
Hence, differentiability is when the slope of the tangent line equals the limit of the function at a given point. This directly suggests that for a function to be differentiable, it must be continuous, and its derivative must be continuous as well.
If we are told that \(\lim _{h \rightarrow 0} \frac{f(3+h)-f(3)}{h}\) fails to exist, then we can conclude that f(x) is not differentiable at x = 3 because it \(f^{\prime}(3)\) doesn't exist.
Now, this leads us to some very important implications — all differentiable functions must therefore be continuous, but not all continuous functions are differentiable!
What?
Simply put, differentiable means the derivative exists at every point in its domain. Consequently, the only way for the derivative to exist is if the function also exists (i.e., is continuous) on its domain. Thus, a differentiable function is also a continuous function.
But just because a function is continuous doesn't mean its derivative (i.e., slope of the line tangent) is defined everywhere in the domain.
How so?
For example, let's look at the graph \(f(x)=|x|\).
We can easily observe that the absolute value graph is continuous as we can draw the graph without picking up your pencil.
Absolute Value – Piecewise Function
But we can also quickly see that the slope of the curve is different on the left as it is on the right. This suggests that the instantaneous rate of change is different at the vertex (i.e., x = 0).
So, what do we do?
We use one-sided limits and our definition of derivative to determine whether or not the slope on the left and right sides are equal.
\begin{equation}
\begin{array}{l}
\lim _{h \rightarrow 0^{-}} \frac{f(x+h)-f(x)}{h}=\lim _{h \rightarrow 0^{-}} \frac{(-(x+h))-(-x)}{h}=\lim _{h \rightarrow 0^{-}} \frac{-x-h+x}{h} \lim _{h \rightarrow 0^{-}} \frac{-h}{\hbar}=\lim _{h \rightarrow 0^{-}}(-1)=-1 \\
\lim _{h \rightarrow 0^{+}} \frac{f(x+h)-f(x)}{h}=\lim _{h \rightarrow 0^{+}} \frac{((x+h))-(x)}{h}=\lim _{h \rightarrow 0^{+}} \frac{x+h-x}{h} \lim _{h \rightarrow 0^{+}} \frac{h}{h}=\lim _{h \rightarrow 0^{+}}(1)=1
\end{array}
\end{equation}
And upon comparison, we find that the slope of the left-side equals -1 and the slope of the right-side equals +1, so they disagree.
Therefore, the function f(x) = |x| is not differentiable at x = 0. While the function is continuous, it is not differentiable because the derivative is not continuous everywhere, as seen in the graphs below.
Derivative Of Absolute Value — Graph
Differentiability Of A Function
So, how do you know if a function is differentiable?
Well, the easiest way to determine differentiability is to look at the graph of the function and check to see that it doesn't contain any of the "problems" that cause the instantaneous rate of change to become undefined, which are:
- Cusp or Corner (sharp turn)
- Discontinuity (jump, point, or infinite)
- Vertical Tangent (undefined slope)
Three Types of Differentiability
So, armed with this knowledge, let's use the graph below to determine what numbers at which f(x) is not differentiable and why.
Ex) What X-Values Is F(X) Not Differentiable?
When Is A Function Not Differentiable
- At x = -8, 0 and 3 (not continuous)
- At x = -4 and 2 (cusp/corner)
- At x = -6.5 (vertical tangent)
See, that's not too difficult to spot, right?
Summary
So, in this video lesson you'll learn how to determine whether a function is differentiable given a graph or using left-hand and right-hand derivatives. In addition, you'll also learn how to find values that will make a function differentiable.
Let's get started!
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Source: https://calcworkshop.com/derivatives/continuity-and-differentiability/