The Alternate Definition of the Derivative

Oct 13, 2025 by pinkandtrash.com 106 views

Introduction

In calculus, the derivative of a function is a measure of how the function changes as its input changes. The derivative is a fundamental concept in mathematics and has numerous applications in various fields, including physics, engineering, and economics. In this article, we will explore the alternate definition of the derivative and use it to determine whether the function $f(x)$ is differentiable at $x = 1$ .

The Alternate Definition of the Derivative

The alternate definition of the derivative is given by:

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

This definition is equivalent to the traditional definition of the derivative, which is given by:

f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}

However, the alternate definition is often more useful when dealing with piecewise functions, as we will see in this article.

The Function $f(x)$

The function $f(x)$ is defined as:

f(x) = \begin{cases} 3 - 2x - x^2 & \text{if } x \leq 1 \\ 5 - 2x & \text{if } x > 1 \end{cases}

This function is a piecewise function, meaning that it is defined differently on different intervals. In this case, the function is defined as a quadratic function for $x \leq 1$ and as a linear function for $x > 1$ .

Determining Differentiability at $x = 1$

To determine whether the function $f(x)$ is differentiable at $x = 1$ , we need to check whether the limit of the difference quotient exists at $x = 1$ . In other words, we need to check whether:

\lim_{h \to 0} \frac{f(1 + h) - f(1)}{h}

exists.

Case 1: $h < 0$

If $h < 0$ , then $1 + h < 1$ , and we have:

f(1 + h) = 3 - 2(1 + h) - (1 + h)^2 = 3 - 2 - 2h - 1 - 2h - h^2 = 0 - 4h - h^2

f(1) = 3 - 2(1) - 1^2 = 0

Substituting these expressions into the difference quotient, we get:

\frac{f(1 + h) - f(1)}{h} = \frac{0 - 4h - h^2 - 0}{h} = -4 - h

As $h \to 0$ , the difference quotient approaches $-4$ , which is a finite value.

Case 2: $h > 0$

If $h > 0$ , then $1 + h > 1$ , and we have:

f(1 + h) = 5 - 2(1 + h) = 3 - 2h

f(1) = 3 - 2(1) - 1^2 = 0

Substituting these expressions into the difference quotient, we get:

\frac{f(1 + h) - f(1)}{h} = \frac{3 - 2h - 0}{h} = 3 - 2h

As $h \to 0$ , the difference quotient approaches $3$ , which is a finite value.

Conclusion

We have shown that the limit of the difference quotient exists at $x = 1$ for both $h < 0$ and $h > 0$ . However, the limit is not the same for both cases. In fact, the limit is $-4$ for $h < 0$ and $3$ for $h > 0$ . This means that the function $f(x)$ is not differentiable at $x = 1$ .

Discussion

The result we obtained is not surprising, given the nature of the function $f(x)$ . The function is a piecewise function, and the two pieces are not "smoothly connected" at $x = 1$ . In fact, the function has a "kink" at $x = 1$ , which makes it difficult to define the derivative at that point.

In conclusion, the alternate definition of the derivative is a powerful tool for determining differentiability of functions. By using this definition, we can show that the function $f(x)$ is not differentiable at $x = 1$ .

References

[1] Thomas, G. B. (2010). Calculus and Analytic Geometry. Addison-Wesley.
[2] Stewart, J. (2012). Calculus: Early Transcendentals. Cengage Learning.

Appendix

The following is a proof of the alternate definition of the derivative.

Theorem

Let $f(x)$ be a function defined on an interval $I$ . Then, the derivative of $f(x)$ at $x = a$ is given by:

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

Proof

Let $f(x)$ be a function defined on an interval $I$ . We need to show that the limit of the difference quotient exists at $x = a$ .

Let $h$ be a small positive number. Then, we have:

\frac{f(a + h) - f(a)}{h} = \frac{f(a + h) - f(a + 0)}{h}

As $h \to 0$ , we have:

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

This shows that the limit of the difference quotient exists at $x = a$ .

The proof for the case $h < 0$ is similar.

Note

Introduction

In our previous article, we explored the alternate definition of the derivative and used it to determine whether the function $f(x)$ is differentiable at $x = 1$ . In this article, we will answer some common questions related to differentiability of functions.

Q: What is differentiability?

A: Differentiability is a concept in calculus that measures how a function changes as its input changes. A function is said to be differentiable at a point if the limit of the difference quotient exists at that point.

Q: What is the difference quotient?

A: The difference quotient is a mathematical expression that is used to calculate the derivative of a function. It is defined as:

\frac{f(x + h) - f(x)}{h}

Q: How do I determine if a function is differentiable at a point?

A: To determine if a function is differentiable at a point, you need to check if the limit of the difference quotient exists at that point. You can do this by using the alternate definition of the derivative, which is:

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

Q: What are some common types of functions that are not differentiable?

A: Some common types of functions that are not differentiable include:

Piecewise functions with "kinks" or "jumps" at certain points
Functions with infinite discontinuities
Functions with vertical asymptotes
Functions with corners or cusps

Q: Can a function be differentiable at a point and still have a discontinuity at that point?

A: Yes, it is possible for a function to be differentiable at a point and still have a discontinuity at that point. For example, the function $f(x) = |x|$ is differentiable at $x = 0$ , but it has a discontinuity at that point.

Q: How do I find the derivative of a function using the alternate definition of the derivative?

A: To find the derivative of a function using the alternate definition of the derivative, you need to follow these steps:

Write down the function and the point at which you want to find the derivative.
Calculate the difference quotient using the formula:

\frac{f(x + h) - f(x)}{h}

Take the limit of the difference quotient as $h$ approaches 0.

Q: What are some common mistakes to avoid when determining differentiability?

A: Some common mistakes to avoid when determining differentiability include:

Not checking if the limit of the difference quotient exists at the point
Not using the correct formula for the difference quotient
Not taking the limit of the difference quotient as $h$ approaches 0
Not considering the possibility of a discontinuity at the point

Conclusion

In this article, we have answered some common questions related to differentiability of functions. We have discussed the concept of differentiability, the difference quotient, and how to determine if a function is differentiable at a point. We have also provided some examples of common types of functions that are not differentiable and some common mistakes to avoid when determining differentiability.

References

[1] Thomas, G. B. (2010). Calculus and Analytic Geometry. Addison-Wesley.
[2] Stewart, J. (2012). Calculus: Early Transcendentals. Cengage Learning.

Appendix

The following is a proof of the alternate definition of the derivative.

Theorem

Let $f(x)$ be a function defined on an interval $I$ . Then, the derivative of $f(x)$ at $x = a$ is given by:

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

Proof

Let $f(x)$ be a function defined on an interval $I$ . We need to show that the limit of the difference quotient exists at $x = a$ .

Let $h$ be a small positive number. Then, we have:

\frac{f(a + h) - f(a)}{h} = \frac{f(a + h) - f(a + 0)}{h}

As $h \to 0$ , we have:

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

This shows that the limit of the difference quotient exists at $x = a$ .

The proof for the case $h < 0$ is similar.

Note

The alternate definition of the derivative is often more useful when dealing with piecewise functions, as we have seen in this article. However, it is worth noting that the traditional definition of the derivative is often more useful when dealing with functions that are defined on a single interval.