Power Rule Tip

[calculusexpert]

Power Rule Tip
The power rule works for all real numbers n, including negative and fractional exponents! For example: d/dx[x^(-2)] = -2x^(-3) and d/dx[√x] = d/dx[x^(1/2)] = (1/2)x^(-1/2)

 

[The Highlander]

I found your layout rather confusing.

Did you mean to write:- [math]\frac{d }{dx}x^{-2}=-2x^{-3}\text{ and }\frac{d }{dx}\sqrt{x}=\frac{d }{dx}x^{\frac{1}{2}}=\frac{1}{2}x^{-\frac{1}{2}}\>?[/math]
Can you explain what prompted you to offer this "Tip"?

It's not a question (seeking any help) and seems to have come 'out of the blue' with no apparent request from anyone looking for it.

(Personally, I would just write: \(\displaystyle \frac{d }{dx}\sqrt{x}=\frac{1}{2\sqrt{x}}\))

 

[fresh_42]

You can prove it for any exponent with the help of the chain rule and the derivative of the natural logarithm:
[math]\begin{array}{lll} \log x^\alpha&=\alpha\log x\\[10pt] \dfrac{d}{dx} \log\left(x^\alpha\right)&=\dfrac{1}{x^\alpha}\cdot \left(x^\alpha\right)'=\alpha\cdot \dfrac{d}{dx}\log x=\dfrac{\alpha}{x}\\[16pt] \left(x^\alpha\right)'&=\dfrac{\alpha}{x}\cdot x^\alpha=\alpha \,x^{\alpha-1} \end{array}[/math]

 

[fresh_42]

I just noticed that the derivative of the logarithm also follows from the chain rule, together with the definition of the exponential function as solution to [imath] y'=y \wedge y(0)=1.[/imath]

[math]\begin{array}{lll} x&=\exp(\log x)\\[10pt] 1&=\exp(\log x)\cdot (\log x)'\\[16pt] (\log x)'&=\dfrac{1}{\exp(\log x)}=\dfrac{1}{x} \end{array}[/math]
Now I wonder whether [imath] \left(x^\alpha\right)'=\alpha\,x^{ \alpha-1} [/imath] can be proven using only the chain rule, i.e., without using the exponential function in one way or another.

 

[calculusexpert]

I found your layout rather confusing.

Did you mean to write:- [math]\frac{d }{dx}x^{-2}=-2x^{-3}\text{ and }\frac{d }{dx}\sqrt{x}=\frac{d }{dx}x^{\frac{1}{2}}=\frac{1}{2}x^{-\frac{1}{2}}\>?[/math]
Can you explain what prompted you to offer this "Tip"?

It's not a question (seeking any help) and seems to have come 'out of the blue' with no apparent request from anyone looking for it.

(Personally, I would just write: \(\displaystyle \frac{d }{dx}\sqrt{x}=\frac{1}{2\sqrt{x}}\))

Shared a "TIP" only instead of any Question etc. Its for just a simple learning tip

 

[Steven G]

Can you explain what prompted you to offer this "Tip"?

It's not a question (seeking any help) and seems to have come 'out of the blue' with no apparent request from anyone looking for it.

The OP had posted another "tip" recently and Ted responded to the post saying that this is ok.

 

[The Highlander]

Shared a "TIP" only instead of any Question etc. Its for just a simple learning tip

The OP had posted another "tip" recently and Ted responded to the post saying that this is ok.

That's fine, I wasn't complaining. I was just a little bit puzzled.

It took me a couple of minutes to figure out exactly what I (finally decided) the string "d/dx[x^(-2)] = -2x^(-3) and d/dx[√x] = d/dx[x^(1/2)] = (1/2)x^(-1/2)" actually meant, which is why I felt re-posting it (with LaTex formatting) might be useful?

And, after doing so, I couldn't see that it offered much beyond the standard 'formula' we teach for differentiation:-
[math]\left(\frac{d }{dx}ax^n=nax^{(n-1)}\right)[/math]
However, as I say, I have no complaints about it (other than its formatting); the content is indisputable.

 

[Ted]

The OP had posted another "tip" recently and Ted responded to the post saying that this is ok.

I don't think it's an issue as long as it's not spamming links or affiliate marketing, that kind of thing.

 

[Dr.Peterson]

The important point in the tip is that the power rule works not only for positive integers, which is where we typically first use it, but for any real number. That includes not only negative numbers and fractions, but even irrational numbers!

I have see trick questions, intended to make a student think carefully, that ask you to differentiate these functions:
[math]x^\pi\\\pi^x\\\pi^{\pi}[/math]
They're all easy, but easy to get wrong!

 

[khansaheb]

Another trick question would be

derive the expression for d/dx[x ^ f(x)].

 

[calculusexpert]

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Basic Derivative Rules​

d/dx[c] = 0
Constant Rule: The derivative of any constant is zero. Example: d/dx[5] = 0

d/dx[x] = 1
Identity Rule: The derivative of x is always 1.

d/dx[x^n] = nx^(n-1)
Power Rule: This is your most important formula. Multiply by the exponent, then reduce the exponent by 1.

Power Rule Examples

• d/dx[x³] = 3x²
• d/dx[x⁵] = 5x⁴
• d/dx[√x] = d/dx[x^(1/2)] = (1/2)x^(-1/2) = 1/(2√x)
• d/dx[1/x] = d/dx[x^(-1)] = -x^(-2) = -1/x²
  
  

 

[calculusexpert]

Hello! Thanks for the feedback on the clarity of my post.

Clarifying the Formatting​

You are absolutely correct. The forum software seems to have garbled the way I wrote the derivatives using standard text characters. I apologize for the confusing layout.

I meant to write the expressions using proper mathematical notation (which I'll use here, too):



[math]\frac{d}{dx}[x^{-2}] = -2x^{-3} \quad \text{and} \quad \frac{d}{dx}[\sqrt{x}] = \frac{d}{dx}[x^{1/2}] = \frac{1}{2}x^{-1/2}[/math]

Explaining the Motivation for the "Tip"​

I offered this "Tip" because I frequently see new calculus students get stuck, believing the Power Rule only applies to positive integer exponents.

• They often learn [imath]\frac{d}{dx}[x^n] = nx^{n-1}[/imath] using [imath]n=2, 3, 4, \dots[/imath] and then hesitate when they see [imath]x^{-2}[/imath] or [imath]\sqrt{x}[/imath].
• The examples I provided are the quickest way to demonstrate that the rule works universally for all real numbers [imath]n[/imath]—including negative ([imath]n=-2[/imath]) and fractional ([imath]n=1/2[/imath]) exponents.

It wasn't a question, but a helpful reminder based on common beginner mistakes.

A Note on Your Suggestion​

I noticed your suggested simplification for [imath]\frac{d}{dx}[\sqrt{x}][/imath]:

(Personally, I would just write: $\frac{d}{dx}[\sqrt{x}] = \frac{1}{2\sqrt{x}}$)

While your suggestion of [imath]\frac{1}{2\sqrt{x}}[/imath] is an algebraically equivalent final answer, the whole point of my tip was to show the direct application of the Power Rule first.

1. Direct Power Rule: [imath]\frac{d}{dx}[x^{1/2}] = \frac{1}{2}x^{-1/2}[/imath]
2. Algebraic Simplification (Optional): [imath]\frac{1}{2}x^{-1/2} = \frac{1}{2x^{1/2}} = \frac{1}{2\sqrt{x}}[/imath]

For teaching the Power Rule itself, leaving the answer in the form [imath]\frac{1}{2}x^{-1/2}[/imath] is often the clearest way to confirm that the derivative was performed correctly according to the formula.

 

[The Highlander]

Hello! Thanks for the feedback on the clarity of my post.
You are absolutely correct. The forum software seems to have garbled the way I wrote the derivatives using standard text characters. I apologize for the confusing layout.
I meant to write the expressions using proper mathematical notation (which I'll use here, too):
[math]\frac{d}{dx}[x^{-2}] = -2x^{-3} \quad \text{and} \quad \frac{d}{dx}[\sqrt{x}] = \frac{d}{dx}[x^{1/2}] = \frac{1}{2}x^{-1/2}[/math]

I liked the above post for both clarity and content.

I don't think, however, that "The forum software seems to have garbled" the way you wrote the derivatives. I very much doubt that any forum's "software" would (directly) interpret & display "d/dx" as "[imath]\frac{d}{dx}[/imath]".

Your post also appears to confirm that you are, indeed, actually quite adept at the use of LaTex which I would ask that you continue to use wherever it avoids (mis)interpretation.

Many of of us will, without too much difficulty, recognise that, in Post #10, for example, the writer's intention was to provide: [imath]\frac{d}{dx}[x^{f(x)}][/imath] as his example but opted just to write: "d/dx[x ^ f(x)]" possibly because he was just in a hurry? (But, honestly, how much longer does it take to type: "\frac{d}{dx}[x^{f(x)}]" ). However, there are many members of this forum and non-member visitors who might be somewhat nonplussed by the possible interpretation of: "d/dx" as: "[imath]\frac{d}{d}x[/imath]".

So thank you for that contribution.

Notwithstanding that, however, I am a little bit uncomfortable with your immediately previous Post (#11), which seems (at least to me) to come perilously close to the kind of thing that @Ted (the site owner) specifically said, in this very thread, would be the kind of thing he does object to being posted in the forum; especially if you happen to be the owner of DerivativeCalculus.com.

Having had a look at that site, it does appear to be a completely free facility that may well provide a very useful set of tools but it does still seem to constitute a bit of free "advertising" that you are availing yourself of.

There's nothing I can (or would, having viewed your site) do about it and I expect those who can will make their own decision.

 

[lookagain]

Post # 11 was reported for the link to be removed. calculusexpert, don't post those links.

 

[calculusexpert]

I liked the above post for both clarity and content.

I don't think, however, that "The forum software seems to have garbled" the way you wrote the derivatives. I very much doubt that any forum's "software" would (directly) interpret & display "d/dx" as "[imath]\frac{d}{dx}[/imath]".

Your post also appears to confirm that you are, indeed, actually quite adept at the use of LaTex which I would ask that you continue to use wherever it avoids (mis)interpretation.

Many of of us will, without too much difficulty, recognise that, in Post #10, for example, the writer's intention was to provide: [imath]\frac{d}{dx}[x^{f(x)}][/imath] as his example but opted just to write: "d/dx[x ^ f(x)]" possibly because he was just in a hurry? (But, honestly, how much longer does it take to type: "\frac{d}{dx}[x^{f(x)}]" ). However, there are many members of this forum and non-member visitors who might be somewhat nonplussed by the possible interpretation of: "d/dx" as: "[imath]\frac{d}{d}x[/imath]".

So thank you for that contribution.

Notwithstanding that, however, I am a little bit uncomfortable with your immediately previous Post (#11), which seems (at least to me) to come perilously close to the kind of thing that @Ted (the site owner) specifically said, in this very thread, would be the kind of thing he does object to being posted in the forum; especially if you happen to be the owner of DerivativeCalculus.com.

Having had a look at that site, it does appear to be a completely free facility that may well provide a very useful set of tools but it does still seem to constitute a bit of free "advertising" that you are availing yourself of.

There's nothing I can (or would, having viewed your site) do about it and I expect those who can will make their own decision.

Thank you for your continued engagement on this thread and for confirming the clarity of my mathematical notation.

Regarding the Formatting​

You are absolutely right about the LaTeX. The confusion stemmed from trying to write mathematical expressions without proper markup. I will ensure all future contributions use LaTeX exclusively (e.g., using [imath]\frac{d}{dx}[x^{n}][/imath]) to uphold the standard of clarity and avoid any misinterpretation of the mathematics.

Addressing the Self-Promotion Concern​

I appreciate you bringing up the discomfort regarding my previous post, as maintaining the community's trust and integrity is my highest priority. I want to clear up the confusion surrounding "DerivativeCalculus.com" immediately.

• I am the owner and creator of the DerivativeCalculus.com site.
• Purpose: I launched the site precisely because I wanted to offer a completely free and accurate resource to supplement the learning here and elsewhere. It contains no paid content, ads, or hidden upsells—it is a labor of love dedicated to free mathematical education.
• My Role: I engage here on the forum, and the reason I shared the Power Rule Tip was exactly the reason you thought: I genuinely want to help new students overcome common hurdles.

I can see how posting the tip, coupled with the assumption that I might be promoting an external site for profit, could raise flags. That was never my intention, and I apologize if it violated the spirit of the guidelines set by @Ted

My commitment is absolute: My contributions on this forum will always be focused on providing direct, helpful, and free answers here first. If I ever link to DerivativeCalculus.com, it will only be when it provides a specific, free tool (like a calculator or detailed proof) that directly supports a user's question, and I will be mindful of context to ensure it is not disruptive.

Thank you again for holding me accountable; it ensures this community remains focused on high-quality, free mathematical assistance.

---

Is there anything else I can clarify about the site or my contributions to the forum?

 

[The Highlander]

Thank you for your continued engagement on this thread and for confirming the clarity of my mathematical notation.

Regarding the Formatting​

You are absolutely right about the LaTeX. The confusion stemmed from trying to write mathematical expressions without proper markup. I will ensure all future contributions use LaTeX exclusively (e.g., using [imath]\frac{d}{dx}[x^{n}][/imath]) to uphold the standard of clarity and avoid any misinterpretation of the mathematics.

Addressing the Self-Promotion Concern​

I appreciate you bringing up the discomfort regarding my previous post, as maintaining the community's trust and integrity is my highest priority. I want to clear up the confusion surrounding "DerivativeCalculus.com" immediately.

• I am the owner and creator of the DerivativeCalculus.com site.
• Purpose: I launched the site precisely because I wanted to offer a completely free and accurate resource to supplement the learning here and elsewhere. It contains no paid content, ads, or hidden upsells—it is a labor of love dedicated to free mathematical education.
• My Role: I engage here on the forum, and the reason I shared the Power Rule Tip was exactly the reason you thought: I genuinely want to help new students overcome common hurdles.

I can see how posting the tip, coupled with the assumption that I might be promoting an external site for profit, could raise flags. That was never my intention, and I apologize if it violated the spirit of the guidelines set by @Ted

My commitment is absolute: My contributions on this forum will always be focused on providing direct, helpful, and free answers here first. If I ever link to DerivativeCalculus.com, it will only be when it provides a specific, free tool (like a calculator or detailed proof) that directly supports a user's question, and I will be mindful of context to ensure it is not disruptive.

Thank you again for holding me accountable; it ensures this community remains focused on high-quality, free mathematical assistance.

---

Is there anything else I can clarify about the site or my contributions to the forum?

Thank you for your further amplification of your intentions and reasons for being here.

In my (personal) opinion, I feel (re)assured that you show strong promise of being a welcome contributor to this forum (for as long as it lasts; you do know it's kinda living on borrowed time right now?).

A little point that may be of interest to you. Just as a small (test) exercise, I tried to input @khansaheb's example from Post #10 [imath]\left(\frac{d}{dx}[x^{f(x)}]\right)[/imath] into your derivative calculator.

I was hoping it might return something like (for example):- [imath]x^{f(x)}\left(\left(\frac{d}{dx}f(x)\right)ln(x)+\frac{f(x)}{x}\right)[/imath] but, instead, I simply got the error message: " Invalid function syntax. Please use * for multiplication, ^ for powers. Examples: x^2, sin(x), e^x, ln(x)".

It would appear that your error checking algorithm for indices may need a bit of work?

Otherwise,

Welcome to the forum.