Problem 55 - Thomas Calculus

Question

Derivative of \(-f\) Does knowing that a function \(f(x)\) is differentiable at \(x=x_{0}\) tell you anything about the differentiability of the function \(-f\) at \(x=x_{0} ?\) Give reasons for your answer.

Step-by-Step Solution

Answer

Yes, \(-f\) is differentiable at \(x_0\) if \(f\) is, with derivative \(-f'(x_0)\).

Step 1: Understand Differentiability

A function is differentiable at a point if its derivative exists at that point. Formally, a function \( f(x) \) is differentiable at \( x = x_0 \) if the limit \( \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h} \) exists.

Step 2: Apply the Concept to the Negative Function

If \( f(x) \) is differentiable at \( x = x_0 \), then by definition \( \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h} \) exists. Consider the function \( -f(x) \). Its difference quotient at \( x = x_0 \) is \( \lim_{h \to 0} \frac{-f(x_0 + h) + f(x_0)}{h} \).

Step 3: Simplify the Difference Quotient

The expression for the derivative of \( -f(x) \) can be rewritten as \( \lim_{h \to 0} \frac{-(f(x_0 + h) - f(x_0))}{h} \). This can be further simplified to \( -\lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h} \).

Step 4: Relate to the Derivative of \( f(x) \)

Since \( \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h} = f'(x_0) \), it follows that \( \lim_{h \to 0} \frac{-(f(x_0 + h) - f(x_0))}{h} = -f'(x_0) \). This indicates that the derivative of \( -f(x) \) at \( x = x_0 \) is \( -f'(x_0) \).

Step 5: Conclusion

The function \( -f(x) \) is differentiable at \( x = x_0 \) if and only if \( f(x) \) is differentiable at \( x = x_0 \). The derivative of \( -f(x) \) is the negative of the derivative of \( f(x) \).

Key Concepts

Derivative

Differentiability is a fundamental concept in calculus, representing the rate at which a function changes at a particular point. A function is said to be differentiable at a point if its derivative exists.
The derivative of a function, denoted as \( f'(x) \), measures how the function's output value changes as its input changes at a specific location \( x \). This is mathematically defined as:

• \( f'(x_0) = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h} \)

The existence of this limit means that as \( h \), which represents a small increment of \( x \), approaches zero, the function's behavior becomes predictable and smooth around \( x = x_0 \).
This property of having a derivative confirms that the function behaves nicely, without jumps or sharp points, at that location.

Difference Quotient

The difference quotient is a significant tool in calculus used to compute the derivative of a function. It represents the average rate of change of the function over a small interval, providing a glimpse into how the function behaves locally.
For a function \( f(x) \), the difference quotient at a point \( x = x_0 \) is given by:

• \( \frac{f(x_0 + h) - f(x_0)}{h} \)

Here, \( h \) is a small change approaching zero. As \( h \) gets smaller, this quotient approximates the function's slope at \( x_0 \), and its limit as \( h \to 0 \) becomes the derivative. This is why the difference quotient is essentially a stepping stone to understanding derivatives—because it expresses precisely how the derivative is calculated.

Negative Function

A negative function simply takes each output value of a given function and multiplies it by \(-1\). This transformation doesn't affect the overall differentiability, though it does change the shape of the graph.
When considering if \(-f(x)\) is differentiable at a point \( x = x_0 \), knowing that \( f(x) \) is differentiable is crucial. If \( f(x) \) has a derivative, then \(-f(x)\) will too. The derivative of \(-f(x)\) is just the negative of the derivative of \( f(x) \):

• \( -f'(x_0) \)

This relation is derived from the fact that multiplying the entire function by \(-1\) essentially multiplies the entire output of the limit process by \(-1\). Thus, all properties of differentiability are maintained, with only the sign changing. This insight keeps the understanding of negative functions simple while connecting beautifully with how derivatives are calculated.