Demonstrating the Differentiability of the Function ( f(x) x^3 ) at ( x 0 )

The concept of differentiability is a fundamental aspect of calculus. It allows us to understand the smoothness of a function at a specific point by examining the existence of the derivative. Here, we will explore the differentiability of the function ( f(x) x^3 ) at the point ( x 0 ).

Definition and Step-by-Step Approach

To determine if the function ( f(x) x^3 ) is differentiable at ( x 0 ), we utilize the definition of the derivative at that point. The derivative ( f'(0) ) is defined by the limit:

[ f'(0) lim_{h to 0} frac{f(h) - f(0)}{h} ]

Let's break down the steps to evaluate this limit:

Evaluating ( f(0) )

First, we evaluate the function at ( x 0 ):

[ f(0) 0^3 0 ]

Substituting into the Derivative Formula

Now we substitute this into the derivative formula:

[ f'(0) lim_{h to 0} frac{f(h) - f(0)}{h} lim_{h to 0} frac{h^3 - 0}{h} ]

This simplifies to:

[ f'(0) lim_{h to 0} frac{h^3}{h} lim_{h to 0} h^2 ]

Evaluating the Limit

As ( h ) approaches 0, ( h^2 ) approaches 0. Thus:

[ f'(0) 0 ]

Since the limit exists, the function ( f(x) x^3 ) is differentiable at ( x 0 ), and the derivative at that point is ( f'(0) 0 ).

Validation Using Limits

As per the definition of differentiability at a point, the right-hand limit should be equal to the left-hand limit. In this case:

[ lim_{h to 0^-} f'(x) lim_{h to 0^ } f'(x) ]

Since the limit evaluates to 0 in both directions, we confirm that the function is differentiable at ( x 0 ).

Using the Formal Definition of the Derivative

We can also verify the differentiability by directly evaluating the derivative:

[ f'(x) lim_{h to 0} frac{f(x h) - f(x)}{h} ]

For ( f(x) x^3 ):

[ f'(x) lim_{h to 0} frac{(x h)^3 - x^3}{h} ]

Expanding and simplifying the expression:

[ f'(x) lim_{h to 0} frac{x^3 3x^2h 3xh^2 h^3 - x^3}{h} lim_{h to 0} frac{3x^2h 3xh^2 h^3}{h} ]

This further simplifies to:

[ f'(x) lim_{h to 0} (3x^2 3xh h^2) 3x^2 ]

Evaluating at ( x 0 ):

[ f'(0) 3(0)^2 0 ]

Thus, the derivative exists and is 0, confirming the differentiability of ( f(x) x^3 ) at ( x 0 ).

Visual Interpretation and Tangent Line

Visually, the differentiability of ( f(x) x^3 ) at ( x 0 ) means that there is a well-defined tangent line at the origin. In this case, the tangent line is the x-axis itself, as the linear approximation of ( f(x) ) at ( x 0 ) is the line ( g(x) 0 ) for all ( x ).

This linear approximation is a valid tangent at the point of tangency, indicating that the function is differentiable at ( x 0 ).

Conclusion

Based on the limit evaluations and the visual interpretation, we conclude that the function ( f(x) x^3 ) is indeed differentiable at ( x 0 ).

Therefore, the answer to the question 'Is ( x^3 ) differentiable at ( x 0 )?' is Yes.