Mastering Differentiability- A Comprehensive Guide to Checking the Differentiability of Functions

How to Check Differentiability

Differentiability is a crucial concept in calculus that determines whether a function is smooth and has a well-defined derivative at every point in its domain. Checking the differentiability of a function is essential for various applications in mathematics, physics, and engineering. In this article, we will discuss different methods to check the differentiability of a function and provide some examples to illustrate the process.

One of the most straightforward ways to check the differentiability of a function is by using the definition of the derivative. According to the definition, a function f(x) is differentiable at a point x = a if the limit of the difference quotient exists as h approaches zero. Mathematically, this can be expressed as:

lim (h → 0) [f(a + h) – f(a)] / h

If this limit exists, then the function is differentiable at x = a. To apply this method, we need to evaluate the limit for each point in the domain of the function. If the limit exists for all points in the domain, then the function is differentiable.

Another method to check the differentiability of a function is by examining its graph. If the graph of a function has sharp corners, vertical tangents, or infinite discontinuities, then the function is not differentiable at those points. However, this method is not always reliable, as it may be difficult to determine the differentiability of a function at a point based solely on its graph.

A more reliable method is to use the rules of differentiation. If a function can be expressed as a combination of differentiable functions, then it is also differentiable. For example, if f(x) = g(x) + h(x), where g(x) and h(x) are differentiable functions, then f(x) is also differentiable.

Let’s consider an example to illustrate the process of checking differentiability. Suppose we have the function f(x) = x^2 sin(x). To check if this function is differentiable, we can use the product rule of differentiation. The product rule states that if f(x) = g(x) h(x), then f'(x) = g'(x) h(x) + g(x) h'(x).

In our example, g(x) = x^2 and h(x) = sin(x). We can find the derivatives of g(x) and h(x) as follows:

g'(x) = 2x
h'(x) = cos(x)

Now, we can apply the product rule to find the derivative of f(x):

f'(x) = g'(x) h(x) + g(x) h'(x)
f'(x) = 2x sin(x) + x^2 cos(x)

Since the derivative of f(x) exists for all real numbers, we can conclude that the function f(x) = x^2 sin(x) is differentiable on its entire domain.

In conclusion, checking the differentiability of a function involves using the definition of the derivative, examining the graph, and applying the rules of differentiation. By understanding these methods, we can determine whether a function is differentiable and analyze its behavior in various applications.