91Ó°ÊÓ

To determine if a function is differentiable at a particular point, we use the concept of the limit of the difference quotient. This method helps us understand how the function behaves around that point. The formula is given by \[ f'(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h} \]Here, \( f'(x) \) represents the derivative, which shows the function's rate of change at \( x \). The difference quotient \( \frac{f(x+h) - f(x)}{h} \) measures the average rate of change over the small interval \( h \).
As \( h \) approaches zero, this average rate of change becomes an instantaneous rate of change if the limit exists.
It's crucial because finding the limit as \( h \to 0 \) tells us if the function smoothly connects over that point.

For functions that have different rules on their left and right, as in piecewise functions, the left-hand derivative becomes significant. It is the derivative calculated when approaching the given point from the left. For instance, for the function given, if \( x < 0 \), we use the left-hand derivative calculated by: \[ \lim_{{h \to 0^-}} \frac{f(h) - f(0)}{h} \]
In our case, this translates to, \( f(x) = (x+1)^2 \), simplifying our expression to \[ \lim_{{h \to 0^-}} \frac{((h+1)^2 - 1)}{h} \].
After simplification, we find it to be \( h + 2 \) as \( h \to 0^- \).
Evaluating this limit as \( h \) approaches zero from the negative side results in \( 2 \), which is the left-hand derivative at \( x = 0 \).
This indicates how quickly or slowly the function is changing from the left.

The right-hand derivative considers the approach from the right of the given point. It's used to measure how the function behaves for \( x \geq 0 \) in this context. For our function, this part is simpler due to its linearity: \[ f(x) = 2x + 1 \]
The derivative of a linear function is constant, reflected as \( 2 \).
To ensure accuracy, we calculate using the limit \[ \lim_{{h \to 0^+}} \frac{f(h) - f(0)}{h} \]
For this, the expression becomes \[ \lim_{{h \to 0^+}} \frac{2h + 1 - 1}{h} = 2 \].
Therefore, as \( h \to 0^+ \), the limit confirms the right-hand derivative as \( 2 \).
This computation indicates how smoothly the function progresses from the right to the point \( x=0 \).

The existence of the derivative at a point signifies that the function is well-behaved and continuous there. For it to exist, both left-hand and right-hand derivatives must be equal.

• This equilibrium ensures that as you approach the point from either direction, the function's rate of change stabilizes to a single value.

In the example provided, we calculate both left-hand (\[ \lim_{{h \to 0^-}} \]) and right-hand (\[ \lim_{{h \to 0^+}} \]) derivatives and find them equal, each being \( 2 \).
Since these derivatives, verified through limits, meet at the point, the derivative \( f'(0) \) indeed exists and is \( 2 \).
Consequently, the function is differentiable at \( x = 0 \). This differentiability ensures that the function transitions without a break or cusp at that point.