# How to determine if a function is differentiable at a point?

Differentiability is a very important concept of calculus that finds its applications in many fields of engineering and science. We can determine whether a function is differentiable or not and hence predict its nature.

## Answer: We can determine if a function is differentiable at a point or not by looking at its graph. A function is not differentiable if its graph has a vertical line, a sharp edge, or discontinuity.

Let's understand the answer in detail.

**Explanation:**

We can determine if a function is differentiable at a point by using the formula: lim h→0 [(f(x + h) − f(x)) / h].

If the limit exists for a particular x, then the function f(x) is differentiable at x.

We can also tell if a function is differentiable by looking at its graph.

A function is not differentiable at a point if:

- The function has a sharp edge at that point.
- The function has a discontinuity at that point.
- The function has a vertical line at that point.

Let's understand with the help of an example.

Let's check if f(x) = x^{1/3} is differentiable at x = 0.

Here, we can see that the graph has a vertical line at x = 0. Hence, it is not differentiable at x = 0.

Now let's check it using the formula:

⇒ lim h→0 [((x + h)^{1/3} − x^{1/3})) / h]

⇒ lim h→0 [((0 + h)^{1/3} − 0^{1/3})) / h]

⇒ lim h→0 (1 / h^{2/3})

Here, As h tends to zero, the denominator is small, so h grows without any bounds, i.e, the graph is vertical. Hence, f is not differentiable at x = 0.

### Hence, we can determine if a function is differentiable at a point or not by looking at its graph. A function is not differentiable if its graph has a vertical line, a sharp edge, or discontinuity.

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