We want differentiable to have the following properties as a bare minimum. However, I think the mistake you have made is to confuse the "left hand derivative" and the "limit of the derivative from the left. These are quite different, as you've discovered. Clearly the "limit of the derivative from the left" is not a good definition of derivative. However, it turns out that the difference quotient makes for a decent definition. So this makes us happy. Otherwise, the left and right limit would disagree.
So this makes us happy as well. Then there's the tangent definition. This one is slightly more troublesome, as you'd have to define tangent without mentioning derivative. This is possible, but what does it buy you? What most people do is define tangent in terms of derivatives! To address your last question, if a function is not defined at a point, how could it be anything at that point? Continuity is not defined outside the domain of a function.
This definition can be extended naturally to multivariate function. The first definition is equivalent to this one because for this limit to exist, the two limits from left and right should exist and should be equal. But I would say stick to this definition for now as it's simpler for beginners. The second definition is not rigorous, it is quite sloppy to say the least.
Also, there's a theorem stating that: if a function is differentiable at a point x, then it's also continuous at the point x. Sign up to join this community.
The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. This directly suggests that for a function to be differentiable, it must be continuous, and its derivative must be continuous as well. Now, this leads us to some very important implications — all differentiable functions must therefore be continuous, but not all continuous functions are differentiable!
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. Thus, a differentiable function is also a continuous function. We can easily observe that the absolute value graph is continuous as we can draw the graph without picking up your pencil. But we can also quickly see that the slope of the curve is different on the left as it is on the right.
A function is said to be differentiable if the derivative of the function exists at all points in its domain. Let us look at some examples of polynomial and transcendental functions that are differentiable:. If f, g are differentiable functions, then we can use some rules to determine the derivatives of their sum, difference, product and quotient.
Here are some differentiability formulas used to find the derivatives of a differentiable function:. In calculus, differentiation of differentiable functions is a mathematical process of determining the rate of change of the functions with respect to the variable. Some common differentiability formulas that we use to solve various mathematical problems are:.
There is an alternative way to determine if a function f x is differentiable using the limits. Let's see the behavior of the function as h becomes closer to 0 from the negative x - axis. What happens when h approaches 0 from right? Now, let's see the behavior of the function as h becomes closer to 0 from the positive x - axis.
We say that a function is continuous at a point if its graph is unbroken at that point. A differentiable function is always a continuous function but a continuous function is not necessarily differentiable. We already discussed the differentiability of the absolute value function. Clearly, there are no breaks in the graph of the absolute value function.
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