Exploring Differentiability: Understanding Smoothness and Derivatives

Topics: Calculus

Differentiability

Differentiability is a measure of how smooth a function is. A function is differentiable at a point if its derivative exists at that point and is non-zero. The graph of a differentiable function is infinitely thin in the sense that it can be covered by an arbitrarily small rectangle. A simple way to think about differentiability is as follows: If you draw two tangent lines to a curve at two different points, and then draw all the possible curves between these two tangent lines, then only those curves that pass through common points on both of your tangent lines will intersect with both of them.

If this intersection happens at any point other than a maxima or minima, then that curve is said to be differentiable at that point (and hence everywhere else). Let’s look at the graph of f(x) = |x. We recall that this function looks like this when x is between -1 and 1. Now we ask, what is the slope of this graph at x = 0? Some of you might think, Oh, it’s 0.

I turn around at that spot. Some of you might be thinking that the slope of this side is negative 1.

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Some of you might be thinking that this side has a slope of 1, because it has a positive slope. Now let’s graph f(x) and see what the derivative is. For any point on the graph to the left of x equals 0, the slope is negative 1. For all values to the left of 0, the derivative is negative 1. To the right of 0, the slope is positive 1. The graph would look like this. Now you can see the next step. The derivative f prime is not iS continuous at 0. If I had asked the question, “What is the limit as x approaches 0 from the left of f(x)?” the answer would 1. be negative 1. But the limit as x approaches 0 from the right of f(x) is positive 1. Thus, the limit as x approaches 0 of f prime of x does not exist because the left and right-hand limits do not agree. The derivative of 0 with respect to itself does not exist. If a function is not differentiable at a point, we say that the function has failed to be differentiable at that point. The derivative of the function is not defined for that particular value of x. The function is continuous, but it is not differentiable. A function is not differentiable at a cusp, or sharp turn. Let’s examine another function. The function g of x is given by g of x equals the cube root of x. You may remember that this function looks like this. The derivative should resemble the following equation. For large values of x, the slope is practically 0, so we would look at a graph of the function and see that the slope is always positive. As the x values approach 0, the slope of the graph approaches a steep slope. Then, after O, our slopes are very steep. Thus, we see that as we approach 0 for this graph of g(x) we get positive infinity. Since infinity is not a finite value, the derivative of f(x) is not defined for x=0, so we can say that f(x) is not differentiable at x=0. If a function is differentiable, then it can be written as a quotient of two functions, one of which is continuous. It is not possible to have a derivative that exists if the function does not exist at a certain point. Imagine a function that is not continuous. How can you draw a line tangent to the curve at this point? You can’t. First, if a function is differentiable, it is continuous. Second, if a function is not continuous, the function will not be differentiable. Those are two main points. Thus, to be differentiable or to show that a function is differentiable, you have to show that f(x) is continuous. When proving that a function has derivative f(x), you first use your definition of derivative and then show that the left-hand derivative is equal to the right-hand derivative. That’s what it is. You would typically test for differentiability at critical points to see if a function is differentiable there. We need functions to be differentiable, because we need derivatives of those functions. If a derivative doesn’t exist, obviously, we can’t find the derivative. Differentiability implies continuity, while a lack of continuity implies that a function is not differentiable. [Graph][Graph][Graph][Graph]