A duty is formally taken into consideration differentiable if that derivative exists at each suggest in that is domain, however what does this mean?

It way that a role is differentiable anywhere its derivative is defined.

So, as long as you have the right to evaluate the derivative at every point on the curve, the function is differentiable.

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How To identify Differentiability

By utilizing limits and also continuity!

The meaning of differentiability is expressed together follows:

f is differentiable top top an open up interval (a,b) if (lim _h ightarrow 0 fracf(c+h)-f(c)h) exists because that every c in (a,b).f is differentiable, definition (f^prime(c)) exists, then f is consistent at c.

Hence, differentiability is as soon as the slope of the tangent line amounts to the border of the role at a provided point. This directly suggests that because that a function to be differentiable, it need to be continuous, and also its derivative should be continuous as well.

If we room told the (lim _h ightarrow 0 fracf(3+h)-f(3)h) fails to exist, then we can conclude the f(x) is no differentiable in ~ x = 3 due to the fact that it (f^prime(3)) no exist.

Now, this leads us to some very important ramifications — all differentiable functions must because of this be continuous, yet not all consistent functions are differentiable!

What?

Simply put, differentiable way the derivative exists at every suggest in its domain. Consequently, the only way for the derivative to exist is if the function also exist (i.e., is continuous) top top its domain. Thus, a differentiable duty is also a continuous function.

But just because a role is consistent doesn’t median its derivative (i.e., slope of the line tangent) is identified everywhere in the domain.

How so?

For example, stop look at the graph (f(x)=|x|).

We can easily observe that the absolute value graph is constant as us can attract the graph without picking up her pencil.


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Absolute worth – Piecewise Function


But us can also quickly see that the steep of the curve is various on the left together it is on the right. This suggests that the instantaneous rate of readjust is various at the crest (i.e., x = 0).

So, what carry out we do?

We use one-sided limits and also our definition of derivative to recognize whether or not the slope on the left and right sides space equal.

eginequationeginarrayllim _h ightarrow 0^- fracf(x+h)-f(x)h=lim _h ightarrow 0^- frac(-(x+h))-(-x)h=lim _h ightarrow 0^- frac-x-h+xh lim _h ightarrow 0^- frac-hhbar=lim _h ightarrow 0^-(-1)=-1 \lim _h ightarrow 0^+ fracf(x+h)-f(x)h=lim _h ightarrow 0^+ frac((x+h))-(x)h=lim _h ightarrow 0^+ fracx+h-xh lim _h ightarrow 0^+ frachh=lim _h ightarrow 0^+(1)=1endarrayendequation

And ~ above comparison, we find that the steep of the left-side equates to -1 and also the steep of the right-side equates to +1, so they disagree.

Therefore, the function f(x) = |x| is no differentiable in ~ x = 0. When the function is continuous, it is no differentiable because the derivative is not constant everywhere, as checked out in the graphs below.


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Derivative the Absolute worth — Graph


Differentiability the A Function

So, exactly how do you know if a role is differentiable?

Well, the easiest method to determine differentiability is come look in ~ the graph that the role and inspect to check out that it no contain any kind of of the “problems” that cause the instantaneous price of change to come to be undefined, i beg your pardon are:

Cusp or edge (sharp turn)Discontinuity (jump, point, or infinite)Vertical Tangent (undefined slope)
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Three types of Differentiability


So, equipped with this knowledge, let’s use the graph listed below to identify what numbers at i m sorry f(x) is not differentiable and also why.

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Ex) What X-Values Is F(X) no Differentiable?


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When Is A role 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 complicated to spot, right?

Summary

So, in this video clip lesson you’ll learn how to identify whether a duty is differentiable provided a graph or using left-hand and right-hand derivatives. In addition, you’ll additionally learn exactly how to discover values that will make a role differentiable.