91Ó°ÊÓ

In calculus, the concept of a derivative is fundamental. It tells us how a function changes at any given point. Think of the derivative as the slope of the tangent line to the graph of the function at a specific point. In simple terms, it measures the rate at which one quantity changes with respect to another. For example, if you know the derivative of a position with respect to time, you know the velocity.

The derivative of a function at a point may be found using the limit definition:

• Given a function \( f(x) \), the derivative \( f'(x) \) is defined as \( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \).

This works for all points where the function is smooth and continuous. However, the situation might become interesting when considering points where the function changes its behavior suddenly, or isn't defined normally. That's where one-sided derivatives come into play.

The right-hand derivative provides insight into how a function behaves as the input approaches from the right, or positive side. It assumes that we are only interested in values greater than or equal to a particular point. For a function \( f(x) \), the right-hand derivative at \( x = a \) is given by:

• \( f^{\prime}(a^{+}) = \lim_{h \to 0^{+}} \frac{f(a+h) - f(a)}{h} \)

This mathematical expression indicates that you are observing the change in \( f \) as \( x \) moves infinitesimally close to \( a \) from the right.

In our specific exercise, for the function \( f(x) = x|x| \), we focused on the expression \( f(x) = x^2 \) for \( x \geq 0 \). By evaluating the derivative using this approach, we found that \( f^{\prime}(0^{+}) = 0 \).

The left-hand derivative helps us understand how a function behaves as the input approaches from the left, or negative side. It examines the behavior of the function for values less than the particular point of interest. The left-hand derivative for a function \( f(x) \) at \( x = a \) is written as:

• \( f^{\prime}(a^{-}) = \lim_{h \to 0^{-}} \frac{f(a+h) - f(a)}{h} \)

By focusing on this formula, we ensure that our understanding of the function encompasses all aspects, particularly as it approaches the specified point from the negative side.

In the case of the function \( f(x) = x|x| \) when \( x < 0 \), we used the expression \( f(x) = -x^2 \). Calculating the derivative in this region, we found \( f^{\prime}(0^{-}) = 0 \).

For a function to be considered differentiable at a specific point, two main criteria must be satisfied:

• The right-hand and left-hand derivatives at that point must exist.
• They must be equal to each other.

Differentiability implies that the function is smooth and has no sharp turns or cusps at that point.

According to our exercise, for the function \( f(x) = x|x| \), we evaluated the right-hand derivative \( f^{\prime}(0^{+}) \) and the left-hand derivative \( f^{\prime}(0^{-}) \). With both being equal to 0, it confirmed that the derivative \( f^{\prime}(0) \) exists and the function is indeed differentiable at \( x = 0 \). Knowing whether or not a function is differentiable at various points is crucial for more advanced studies and applications in calculus.