Understanding the Differentiability of Functions at Specific Points

In this article, we will explore the concept of differentiability of functions, particularly focusing on is the function f(x) differentiable at x0. We will also discuss the differentiability of the square-root function fx √x. By the end of this article, you will have a clear understanding of the conditions under which a function can be considered differentiable.

The Function at x0

The provided function is defined as:

f(x) begin{cases} sqrt{x} x geq 0 sqrt{-x} x 0 end{cases}

Let's first analyze the nature of this function:

The function is not differentiable at x0. This is because the derivative is undefined at this point. We can see this by examining the derivative of the function.

For x 0, the derivative can be computed as:

f'(x) frac{1}{2sqrt{x}}

For x 0, the derivative is:

f'(x) -frac{1}{2sqrt{-x}}

Note that the derivative at x0 is undefined because:

lim_{x to 0} f(x) begin{cases} infty x geq 0 -infty x 0 end{cases}

Not only is the derivative undefined due to being unbounded, but it also gives different values when approached from both sides. This means that the function does not have a well-defined tangent line at x0.

The Differentiability of Square-Root Function

Next, let's consider the function fx √x.

This function is differentiable at all x ≥ 0. To understand why, let's examine its derivative:

f(x) √x

For x 0, the derivative can be computed as:

f'(x) frac{1}{2sqrt{x}}

The function is not differentiable at x 0. This is because the derivative of the square-root function does not exist at the origin, given that you cannot approach that point from both directions. However, this is a minor point since the right-hand limit exists and the graph of fx is approaching a vertical tangency environment.

Conclusion

In summary, the differentiability of a function depends on its behavior at specific points. For the given function f(x), it is not differentiable at x0 due to the unbounded nature of its derivative and the different limits from both sides. For the square-root function, fx √x, it is differentiable at all x ≥ 0, with the exception of x0, where the right-hand limit exists.

Understanding these concepts is crucial in calculus and real analysis, as they help in determining the behavior of functions and their applications in various mathematical and scientific fields.

Keywords: differentiability, function, derivative