What Does Differentiable Mean in Calculus? 

A function is differentiable if there is a derivative for every point on its domain. However, not all functions are differentiable. For example, a function is not differentiable if there is a discontinuity at a point. The exception to this rule is if the function has a corner or a kink. 

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In math, a differentiable function is a smooth continuous function that can be locally approximated by a linear function at a given interior point. As a result, a differentiable function will have a tangent line at each point in its domain. 

A tangent line at a particular point will usually look like a dashed line. This is due to the fact that the slope of the line will be approaching the value of the slope of the tangent line from the opposite side. When the slope of the tangent is close to zero, it is likely to be a straight line. Similarly, the limit at a point will usually be a tangent line. If a point does not have a tangent line, there is no point in trying to calculate its derivative. 

In mathematics, a differentiable function is merely a function that has a derivative at every point in its domain. In practice, most functions have derivatives at nearly every point in their domain. Of course, there are some functions that have a singular or discontinuous derivative at a particular point. There are also some functions that have a non-continuous derivative at a particular point. One of the most well known examples of a continuous function with a non-continuous derivative is the Weierstrass function. 

Differentiable functions aren’t as common as other continuous functions. But they do exist and you can prove their existence with a bit of imagination. It’s important to understand that a function’s derivative cannot be infinitely large. Additionally, there are some anomalies that can cause a function to not be differentiable. Specifically, a function may have a jump, a kink, or a corner. These are all atypical characteristics of a continuous function. 

To prove the existence of a function’s derivative at a point, you need to know the right formula. For instance, a derivative of f is defined by f(x) = x2sin(1/x). What you need to do is to prove that f'(a) = x2sin(1/x) is the same as f'(a)= x2sin(1/x). That is, f'(a)= x2sin(1/x) if a = 0. 

Differentiable functions are a great example of the need for a smooth and relatively simple graph. In the case of a function, this means that it isn’t going to have sharp corners or breaks. Moreover, it is going to have a non-vertical tangent line at each of its interior points. Consequently, the graph must be smooth and relatively free of holes. 

The tangent line will also tell you that a function is differentiable. Specifically, the tangent line at a particular point is a good approximation to a curve when x is close to 0. By the same token, a function will have a tangent line with a sharp bend at a specific point.