Question

Using one-sided derivatives, show that the function \(f(x)=\left\\{\begin{array}{c}{x^{2}+x,} & {x \leq 1} \\ {3 x-2,} & {x>1}\end{array}\right.\) does not have a derivative at \(x=1\)

Short answer

The function seems to be differentiable at x=1 as both left-hand and right-hand derivatives at point x=1 are equal (3). It contradicts the initial task of the exercise.

Step-by-step solution

Compute the Left-Hand Derivative at x=1

The left-hand derivative of the function when x=1, is calculated by taking the derivative of the function for x<=1. This means using the function \( f(x) = x^2 + x \). The derivative is \(2x + 1\). Substitute x=1 into this, the left hand derivative will be \(2(1) + 1 = 3\).

Compute the Right-Hand Derivative at x=1

The right-hand derivative of the function when x=1, is calculated by taking the derivative of the function for x>1. This means using the function \( f(x) = 3x - 2 \). The derivative is 3. Substitute x=1 into this, the right-hand derivative will be 3.

Compare the Left-hand and Right-hand Derivatives

The function has a derivative at x=1 if and only if the left-hand and right-hand derivatives are equal. In this case, the left-hand derivative at x=1 is 3, and the right-hand derivative at x=1 is also 3. Therefore, both derivatives are equal.

Verify and Conclude

The left-hand and right-hand derivatives are both 3, therefore, the function is differentiable at x=1. It means there seems to be a mistake in the exercise or our calculations, because they contradict the initial assumption of the exercise - that there's no derivative at x=1.

Key concepts

Derivative at a Point

The derivative of a function at a specific point measures the rate at which the function's value changes as its input changes. It represents the slope of the tangent line to the function's graph at that point. Mathematically, if we consider a function denoted by \( f(x) \), the derivative at a point \( x = a \) is defined as the limit of the difference quotient as \( h \) approaches zero:
\[ f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \].
However, this definition assumes that the limit exists and is the same as \( h \) approaches zero from both the left and the right. If these one-sided limits aren't equal, the function doesn't have a derivative at that point. This property will be essential when questioning the differentiability of piecewise functions.

Piecewise Functions Calculus

Piecewise functions are defined by different expressions depending on the value of the independent variable. When working with calculus involving piecewise functions, it's critical to consider each piece separately. For these types of functions, the derivative is not necessarily the same across different intervals. One-sided derivatives become particularly useful when evaluating the derivative at points where the function's formula changes.
In piecewise functions, a common challenge is to determine the differentiability at the points of transition from one formula to another. These points are where the definition of the function switches, and they often warrant closer examination to ensure continuity and differentiability.

Continuity and Differentiability

For a function to be differentiable at a point, it must be continuous there. Continuity requires that the left-hand limit, the right-hand limit, and the function's value at the point are all equal. If a function is not continuous at a point, it cannot be differentiable there.
Differentiability is, in a sense, a stronger condition than continuity. A function is differentiable at a point if it is continuous there and if the derivative exists at that point. The existence of the derivative demands that the function's rate of change is the same as we approach the point from either direction. This is not necessarily true for all continuous functions, and that's why continuity doesn't always guarantee differentiability.

Left-Hand and Right-Hand Limits

The concept of one-sided limits addresses the behavior of a function as the input approaches a specific value from one direction—either from the left or the right. The left-hand limit (as \( x \) approaches \( a \) from the left) is denoted as \( \lim_{x \to a^-} f(x) \), while the right-hand limit (as \( x \) approaches \( a \) from the right) is denoted as \( \lim_{x \to a^+} f(x) \).
In the context of derivatives, the one-sided derivatives are the limits of the difference quotient when the input approaches the point exclusively from one side. For a function to have a derivative at a point, both the left-hand derivative and the right-hand derivative must exist and be equal. If they differ or if one does not exist, the function is not differentiable at that point. This distinction is crucial when examining piecewise functions at their breakpoints.