What makes it differentiable

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.

The function is continuous everywhere. Hence the main difference between a differentiable and continuous function is that a differentiable function is always a continuous function but a continuous function may not be differentiable.

Differentiation of Trigonometric Functions. Solution: We will use the quotient rule for differentiable functions to determine the derivative of f x.

Now, let's use the limit definition of differentiable functions. We already observed that the limits are different for absolute value of function. Yes, the cubic function is differentiable. If a function is twice differentiable, then it means that the second derivative of the function exists. The absolute value function is not differentiable at 0 because the graph of the function has a sharp corner at this point.

A function can be proved differentiable if its left-hand limit is equal to the right-hand limit and the derivative exists at each interior point of the domain. Learn Practice Download. Differentiable A differentiable function is a function in one variable in calculus such that its derivative exists at each point in its entire domain. What is Differentiable? Rules for Differentiable Functions 3.

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. 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. While the function is continuous, it is not differentiable because the derivative is not continuous everywhere, as seen in the graphs below.

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Have you ever wondered what makes a function differentiable? Absolute Value — Piecewise Function.