91Ó°ÊÓ

Understanding differentiability is key to mastering calculus. Differentiability refers to the ability of a function to have a derivative at every point within its domain. A function is said to be differentiable at a particular point if the derivative exists at that point and if the function is smooth and continuous there. This smoothness implies that there are no sharp corners or cusps at the point in question.

When we consider the function in the exercise, \(f(x)=x^{3}-4x\), and evaluate its differentiability at \(x=0\), we need to ensure that the function is continuous and smooth at this point. The function provided is a polynomial, which is continuous and smooth everywhere in its domain. Hence, it is differentiable at \(x=0\), as confirmed by calculating the numerical derivative. For functions to be differentiable, they also must be continuous at that point – which leads us to our next concept.

The derivative of a function at a specific point gives us the slope of the tangent line to the curve of the function at that point. This is a powerful concept as it helps us understand the rate of change of the function at an infinitesimally small interval around the point. To numerically find the derivative at a point, we take a small value for \(h\), which represents a minute change in our input, and apply it to the function to see how it affects the output.

In the given exercise, the formula \(f'(x) = (f(x + h) - f(x))/h\) is used with a small value of \(h=0.001\) to approximate the derivative at \(x=0\). The process essentially measures the steepness or the 'slope' of the function at the exact point, providing us a glimpse into the instantaneous rate of change. This finite calculation is an approximation of the derivative, which would technically require \(h\) to approach zero.

Limits and continuity are the foundation upon which differentiability rests. A limit, in calculus, is essentially the value that a function approaches as the input approaches a certain value. Continuity, on the other hand, means that the function is unbroken and has no gaps, jumps, or points of discontinuity in its domain. If a function is not continuous at a point, it cannot be differentiable there either.

In our exercise example, before we can declare a function as differentiable at a certain point, we first have to ensure that it's continuous at that point. The function \(f(x)=x^{3}-4x\) is a polynomial, and polynomials are known to be continuous for all real numbers. This implies that the function has limits that correspond to the actual value of the function at every point, including \(x=0\). Because of this, along with our numerical derivative result, we confidently say that our function is not only continuous but also differentiable at the indicated point.