Polynomial Derivatives: The Power Rule

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Lesson Priority: VIP Knowledge

 
Objectives
  • Learn the Power Rule for quickly taking the derivative of any single variable monomial
  • Take the derivative of entire polynomials by applying the power rule one term at a time
  • Apply the Power Rule for variables raised to fraction exponent powers
  • See and practice using the power rule correctly when the variable is in the denominator
Lesson Description

By far the most common type, polynomial derivatives have a simplistic, formulaic approach that we can use instead of performing the long hand limit definition method of derivatives. We will learn the massive time-saving formula approach in this lesson.

 
Practice Problems

Practice problems and worksheet coming soon!

 

The Most Common Derivative

Now we turn to the most common type of derivative we will take throughout our Calculus journey - one that is both gratifying and infuriating. The gratification comes from the fact that simple derivatives that we had to do via the long song and dance of the limit definition of the derivative now become easy mental math. The infuriating part is the fact that we had to spend all that time doing it the long and complicated way.Many students actually get to learn the power rule polynomial shortcut in advance. It is, after all, the practical and useful takeaway. But most courses still make you learn the stuff we saw in the lesson on finding derivatives by using the limit definition. That's just part of the course.
Define: The Power Rule for DerivativesThe derivative of a variable raised to a numeric exponent is computed as follows.$$\frac{d}{dx} \, x^n = nx^{n-1}$$
Here are some quick examples.
 
Example 1$$x^2 \Rightarrow 2x$$Example 2$$x^{10} \Rightarrow 10x^9$$Example 3$$\sqrt{x} = x^{1/2} \Rightarrow \left(\frac{1}{2}\right) \, x^{-1/2}$$Example 4$$x^{\pi} \Rightarrow \pi x^{\pi-1}$$
 
Compare some of these results with the load of work we had to do using the limit definition - for example:Example 1 (redone)$$\blacktriangleright \,\, \frac{d}{dx}\,\, x^2 = \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h}$$$$=\lim_{h \to 0} \frac{\cancel{x^2} +2xh + h^2 - \cancel{x^2}}{h}$$$$=\lim_{h \to 0} \frac{\cancel{h}\left(2x + h\right)}{\cancel{h}}$$$$=\lim_{h \to 0} 2x + h = \boxed{2x}$$This is the same answer we got using the new shortcut.
Pro Tip
You or someone in your class may actually learn the Power Rule shortcut early on, and discuss it while the class is still learning the limit definition material. Knowing the shortcut even before you learn the limit definition is ok, and sometimes helpful, since you can check your work easily. However, never make the mistake of executing a "limit definition" problem using the shortcut. Failing to do and show the work in that case will usually get you zero credit.
Hopefully the usefulness of the Power Rule is quickly becoming clear. We can now very quickly handle derivatives of typical polynomial terms without any real effort or derivation. We can also extend this knowledge a bit using standard derivative rules that we recently unearthed - together with the Power Rule, those rules allow us to make quick work of all polynomial derivatives.

Linear Operators with the Power Rule

Recall the linear operator derivative rules from the recent lesson on Derivative Rules » - specifically the following two:$$\frac{d}{dx}\,\, c \cdot f(x) = c \cdot \frac{d}{dx} \,\, f(x)$$(1)$$\frac{d}{dx} [f(x) + g(x)] = \frac{d}{dx}\,\, f(x) + \frac{d}{dx}\,\, g(x)$$(2)Equation (1), often called the "constant rule", allows us to apply the Power Rule to polynomial terms that already have coefficients. This is common, and something we should be expertly at by the time we're done practicing - so much that we are handling the computation as mental math.For example, the rules allow us to perform$$\frac{d}{dx}\,\, 4x^3$$as$$4 \cdot \frac{d}{dx}\,\, x^3 = 4 \cdot (3x^2) = 12x^2$$With practice, by the time we are ready to take a quiz on this stuff, we really ought to be able to do this in one step:$$\frac{d}{dx}\,\, 4x^3 = 12x^2$$To do this, focus on what the Power Rule is really telling us to do: move the original exponent (in this case, $3$) down in front as a multiplier. Then decrease the exponent by $1$. Voila!See if you can compute the following several derivatives without any scratch work. Click on each answer to check your work!
 
Examples 5$$\frac{d}{dx}\, 5x^2$$
Show solution
$$\blacktriangleright \,\, 10x$$
 
Example 6$$\frac{d}{dx} \, -x^4$$
Show solution
$$\blacktriangleright \,\, -4x^3$$
 
Example 7$$\frac{d}{dx} \, 6x^7$$
Show solution
$$\blacktriangleright \,\, 42x^6$$
 
Example 8$$\frac{d}{dx} \, \frac{x^6}{2}$$
Show solution
$$\blacktriangleright \,\, 3x^5$$
 
Now that we've seen the direct implication of the constant rule (equation (1) above). The sum and difference rule (equation (2) above) simply allows us to tackle entire polynomials at a time.
 
Example 9$$\frac{d}{dx} \, \big[ x^3 -2x^2 + 6x + 3 \big]$$$\blacktriangleright$ This can be done by computing each term's derivative in turn. Technically we are applying the sum and difference rule stated in equation (2):$$\frac{d}{dx} \, \big[ x^3 -2x^2 + 6x + 3 \big]$$$$= \frac{d}{dx} \, x^3 - \frac{d}{dx} \, 2x^2 + \frac{d}{dx} \, 6x + \frac{d}{dx} 3$$In terms of practicality, no one actually expects us to write each $\frac{d}{dx}$ term out each time. We just go for it. Therefore, by computing the derivative of each term, we can say that$$\frac{d}{dx} \, x^3 -2x^2 + 6x + 3$$$$= 3x^2 - 4x + 6$$

Simple Rational Terms

We commonly associate the Power Rule with polynomials, but it works for any expression in the form $x^n$. Technically, terms can only be called polynomial terms if the power ($n$) is a non-negative integer. However, the Power Rule works for any expression $x^n$ where $n$ is any real number. The first case we'll see of this is for simple rational terms, and the second is for root expressions.
 
Example 10$$\frac{d}{dx} \,\, \frac{1}{x^2}$$$\blacktriangleright$ Before we take the derivative, we can rewrite this expression in the form $x^n$. That way, we can find the derivative by using the Power Rule.$$\frac{d}{dx} \,\, \frac{1}{x^2} = \frac{d}{dx} \,\, x^{-2}$$$$= -2x^{-3} = \frac{-2}{x^3}$$It is important to note that most teachers and profs will not allow you to turn in answers with negative exponents (though some will - but it's rare). Make sure you rewrite your final answer without negative exponents.
Warning!
It is not appropriate to leave negative exponents in your answer, and it is common to lose points if you do so. Just move the variables to or from the denominator as needed to make sure you are not ignoring this requirement.

Fraction Exponents

Taking derivatives of roots of variables or variables raised to fraction exponents is just plain ugly unless you use the Power Rule. This is an absolute necessity and a definite needed skill for quizzes and exams, and very much present for the rest of the course.Let's jump right in with an example:
 
Example 11$$\frac{d}{dx} \,\, \sqrt{x}$$$\blacktriangleright$ Before taking the derivative, rewrite the problem without using the $\sqrt{\, \, \,}$ symbol. This will be the standard procedure for derivative problems that contain radicals.$$\frac{d}{dx} \,\, x^{1/2}$$Again, notice how we specifically took the time to rewrite the expression without a radical symbol. I cannot stress the importance of this step enough. Quiz grades are made or lost on this step. Now we can proceed with using the Power Rule verbatim.$$\frac{d}{dx} \,\, x^{1/2} = \left( \frac{1}{2} \right) x^{-1/2} = \frac{1}{2x^{1/2}}$$Typically there is no need to write the final answer with the radical symbol either - though I have encountered professors who demand it on very rare occasion. According to standard math conventions, $x^{1/2}$ is just as simplified as $\sqrt{x}$.
Remember!
Taking the derivative of a root expression without making a mistake is a difficult task, without rewriting the expression using fraction exponents. Additionally, do not make the common mistake of trying to do too much at once. Every time, rewrite the expression using fraction exponents before you take the derivative. Then, with a cleaner problem to work with, proceed with differentiation.
Here's another example to try.
 
Example 12$$\frac{2}{3\sqrt[4]{x^3}}$$
Show solution
$\blacktriangleright$ First, rewrite the expression using exponent form$$\frac{d}{dx} \,\, \frac{2x^{-3/4}}{3}$$Your scratch work needs to clearly show that the variable, which has a coefficient of $(2/3)$ at the start of the problem, will be multiplied by its current exponent of $(-3/4)$ and end up being raised to a power one unit smaller than its current exponent of $(-3/4)$, or $(-3/4)-(1)=(-7/4)$. All the while, each value needs to be clearly shown as a numerator or denominator object, and the final answer should not have a negative exponent.$$\frac{d}{dx} \frac{2x^{-3/4}}{3} = \left( \frac{2}{3} \right) \, \left( - \frac{3}{4} \right) \, x^{-7/4}$$$$=\frac{6}{12x^{7/4}}=\boxed{\frac{1}{2x^{7/4}}}$$
Pro Tip
With radical or fraction exponent problems, there are potentially a lot of fractions involved in your arithmetic, both with coefficients and exponents. Keep straight which objects are in the denominator, and don't skimp on the parenthesis to keep it obvious to your teacher.

Perils and Pitfalls

While it is impossible to anticipate every type of mistake I have seen students make (or will see students make) on this particular topic, here are a few common ones to look out for.1) Don't leave x in the denominatorThere are very few students who can take the derivative of something like$$\frac{1}{x^2}$$without making a mistake unless first rewriting it as $x^{-2}$. Those that can are usually doing several steps at once in their head. I'm all for lazy scratch work when we can, but you're really not saving many pencil strokes here by doing this. Do yourself a favor and rewrite it before differentiating. You're bound to make fewer mistakes.2) Don't leave x under a rootSimilar to leaving $x$ in the denominator, students who fail to rewrite something like$$\sqrt{x}$$before taking the derivative are much more likely to end up with a wrong answer.3) Don't apply the Power Rule to a ProductThe Power Rule cannot be directly applied to the product of two or more polynomials. The derivative of such an object can be done (in fact you can take the derivative of virtually anything, as you'll come to learn) but the Power Rule only applies to individual terms that are added or subtracted together.
 
Example 13$$\frac{d}{dx} \,\, (2x^2 + x -7)(-5x^3 + 3x^2 - x - 1)$$$\blacktriangleright$ You might be tempted to take the derivative of each polynomial and multiply those results together, but that absolutely does not work. There does exist a generalized rule for finding the derivative of a product, and this aptly named Product Rule is the subject of an upcoming lesson ».In the meantime, the only other way to perform this differentiation is to actually multiply the expression out using the distributive rule, and then proceed as normal with the resulting polynomial. In this case:$$(2x^2 + x -7)(-5x^3 + 3x^2 - x - 1)$$$$ = -10 x^5+x^4+36 x^3-24 x^2+6 x+7$$and the derivative of that result is$$\boxed{-50x^4 + 4x^3 + 108x^2 -48x + 6}$$Notice how if we erroneously attempted to solve this problem by taking the derivative of each polynomial ($4x+1$ and $-15x^2 +6x -1$, respectively) and multiplying them together, we would get$$-60 x^3+9 x^2+2 x-1$$which does not match the correct answer that we have already calculated.4) Don't apply the Power Rule to an expressionAnother object that the Power Rule doesn't exactly work for is entire expressions raised to a power, such as$$(x^2 + 1)^4$$whose derivative is not $4(x^2+1)^3$ (though we'll see via the Chain Rule » that it is close). If we really needed to we could brute-force this problem right now by actually multiplying this expression out to obtain a plain polynomial.$$(x^2 +1)^4 = (x^2+1)(x^2+1)(x^2+1)(x^2+1)$$However, this is almost never advisable for many reasons. We won't proceed that way ever in similar Mr. Math examples, instead opting for the method in the upcoming Chain Rule » lesson. In fact, when teachers test this concept using the forthcoming "Chain Rule", they usually give problems similar to$$(x^2 + 1)^{999}$$which of course, you can not expand manually, unless you're bored serving a 1 year prison sentence. And even then you might not finish before your time is up.

Mr. Math Makes It Mean

A common way that the power rule is used in a cruel way is to give you an irrational exponent - usually $e$ or $\pi$. Because these are just numbers, the power rule applies verbatim. However, most students freeze up and think something is different because these special numbers are being used. It's not. For example:$$\frac{d}{dx} \,\, x^{e} = ex^{e-1}$$$$\frac{d}{dx} \,\, x^{\pi} = \pi x^{\pi-1}$$

Put It To The Test

In this "Put It To The Test" section, we'll see a good mix of what to expect on a test when your class first covers this topic. However, you'll be using this in more complicated future topics as well, so if these problems look easier than the stuff your class is working on, then your current material is probably using concepts we have yet to cover. Browse the lessons on Differentiation Techniques » to see more concepts where several derivative rules start to mix together. Also, do all the practice problems for this lesson in the Mr. Math worksheets! There's a lot of them but they are super quick, and this skill is super important if you like good grades.
 
Example 14$$\frac{d}{dx} \, 3x^7$$
Show solution
$\blacktriangleright$ The coefficient that is already there ($3$) stays where it is, and we follow the Power Rule by bringing down the $7$ exponent as a multiplier, while also reducing the exponent by $1$.$$\frac{d}{dx} \, 3x^7 = 3 \cdot 7 \cdot x^6$$$$=21x^6$$
 
Example 15$$\frac{d}{dx} \, 2x^{-\frac{3}{4}}$$
Show solution
$\blacktriangleright$ The Power Rule still applies, though the coefficients are not all going to be whole numbers. Bring down and reduce just like we should, but also for this one we must make sure to simplify.$$\frac{d}{dx} \, 2x^{-\frac{3}{4}} = 2 \left(-\frac{3}{4}\right) x^{-\frac{3}{4}-1}$$$$= -\frac{3x^{-\frac{7}{4}}}{2}$$
 
Example 16$$\frac{d}{dx} \, -x^4$$
Show solution
$$\blacktriangleright \,\, \frac{d}{dx} \, -x^4 = -4x^3$$This is a straight forward application of the Power Rule.
 
Example 17Find $f'(x)$$$f(x) = x^{3.1}$$
Show solution
$\blacktriangleright$ There is nothing new or different for this problem - it is still a Power Rule problem, since it is of the form $x^n$. But you'll find that teachers love putting funky exponents up there just to try and throw students off. All we need to do it bring the power down as a multiplier and reduce the power by one.$$\frac{d}{dx} \, x^{3.1} = \boxed{3.1x^{2.1}}$$
 
Example 18$$\frac{d}{dx} \, \frac{1}{x^5}$$
Show solution
$\blacktriangleright$ As with any problem that has $x$ raised to a negative or fraction power, we need to do ourselves the favor of re-writing the problem in the classic $x^a$ form.$$\frac{d}{dx} \, \frac{1}{x^5} \longrightarrow \frac{d}{dx} \, x^{-5}$$Now we're back to a straightforward application of the Power Rule.$$\frac{d}{dx} \, x^{-5} = -5x^{-6}$$And just as important as getting the process right, remember that you need to present answers without negative exponents. So our final answer is$$\frac{d}{dx} \, x^{-5} = -\frac{5}{x^{6}}$$
 
Example 19$$\frac{d}{dx} \, \frac{1}{\sqrt[3]{x}}$$
Show solution
$\blacktriangleright$ Just like in the last problem, we need to re-write this in $x^a$ form in order to move forward.$$\frac{d}{dx} \, \frac{1}{\sqrt[3]{x}} \longrightarrow \frac{d}{dx} \,\, x^{-\frac{1}{3}}$$$$=\left(-\frac{1}{3}\right) x^{-\frac{4}{3}}$$And no matter how many fractions or negative signs we find, we must clean it up into a single fraction with no negative exponents!$$\frac{d}{dx} \, \frac{1}{\sqrt[3]{x}} = \boxed{-\frac{1}{3x^{4/3}}}$$
Remember!
Every derivative problem you turn in, in this lesson and in all future ones, must be cleaned up and simplified. You will lose points for breaking form! Make sure you 1) have an answer that is written as a single fraction, if applicable, and 2) contains only positive exponents.
 
Example 20$$\frac{d}{dx} \,\, \big[ 5x^2 +4x - 3 \big]$$
Show solution
$\blacktriangleright$ We will proceed by taking the derivative of each term.$$\frac{d}{dx} \,\, \big[ 5x^2 +4x - 3 \big] = 10x + 4 $$
Remember!
Don't forget that the derivative of a solo constant is zero, unlike constants that are attached to variable terms, which simply stay put.
 
Example 21$$\frac{d}{dx} \, \big[10x^3 + 8x^2 + 100\big]$$
Show solution
$\blacktriangleright$ We will proceed by taking the derivative of each term.$$\frac{d}{dx} \, \big[10x^3 + 8x^2 + 100\big]$$$$=30x^2 + 16x$$
 
Example 22$$\frac{d}{dx} \, \pi x^{\pi}$$
Show solution
$\blacktriangleright$ This is another common example where the teacher will try to trick you. Remember, $\pi$ is just a number!$$\frac{d}{dx} \, \pi x^{\pi} = \pi^2 x^{\pi-1}$$
 
Example 23$$\frac{d}{dx} \, \big[3x^5 - 5\sqrt{x} + \frac{4}{x}\big]$$
Show solution
$\blacktriangleright$ This is yet another case where we won't be able to proceed easily without re-writing each term in the classic $x^a$ form.$$\frac{d}{dx} \, \big[3x^5 - 5x^{\frac{1}{2}} + 4x^{-1}\big]$$$$15x^4 - 5 \cdot \left(\frac{1}{2}\right) x^{-\frac{1}{2}} + (4)(-1)x^{-2}$$$$=15x^4 - \frac{5}{2x^\frac{1}{2}} - \frac{4}{x^2}$$Once again, make sure your answer does not contain negative exponents.
 
Example 24$$\frac{d}{dx} \sqrt{x}\left(x^2 - 2\right)$$
Show solution
$\blacktriangleright$ Make sure you avoid the common pitfall that many students are tempted by - you cannot take the derivative of $\sqrt{x}$ and $\left(x^2 - 2\right)$ separately and simply multiply the results together. You can two options: 1) Use the Product Rule that we haven't learned yet, or 2) multiply the expression outright and then proceed to take the derivative. As you may imagine, we are choosing the latter path.$$\frac{d}{dx} x^{\frac{1}{2}}\left(x^2 - 2\right)$$$$=\frac{d}{dx} x^{\frac{5}{2}} - 2x^{\frac{1}{2}}$$Now we can take the derivative.$$=\left(\frac{5}{2}\right) x^{\frac{3}{2}} - (2)\left(\frac{1}{2}\right)x^{-\frac{1}{2}}$$$$=\frac{5x^{\frac{3}{2}}}{2} - \frac{1}{x^{\frac{1}{2}}}$$
 
Example 25$$\frac{d}{dx} \, \bigg[-2x^5 + 4x^4 - 6x^2 + x\sqrt{2}$$$$ - 2\sqrt{x} + \frac{1}{x} + \frac{2}{x^{3/2}} - \frac{4}{3x^2}\bigg]$$
Show solution
$\blacktriangleright$ This long problem is quite the mixed bag, but as we have been seeing, we need to rewrite this term-by-term before we take the derivative, so that we have a smooth ride.$$\frac{d}{dx} \, \bigg[-2x^5 + 4x^4 - 6x^2 + x\sqrt{2}$$$$- 2x^{\frac{1}{2}} + x^{-1} + 2x^{-\frac{3}{2}} - \left(\frac{4}{3}\right) \cdot x^{-2} \bigg]$$ $$\longrightarrow -10x^4 + 16x^3 -12x + \sqrt{2}$$$$ - (2)\left(\frac{1}{2}\right)x^{-\frac{1}{2}} + -x^{-2}$$$$ - (2)\left(\frac{3}{2}\right) x^{-\frac{5}{2}} + \left(\frac{4}{3}\right) (-2) x^{-3}$$ $$\longrightarrow -10x^4 + 16x^3 -12x + \sqrt{2}$$$$ - \frac{1}{x^{\frac{1}{2}}} - \frac{1}{x^2} - \frac{3}{x^{\frac{5}{2}}} - \frac{8}{3x^3}$$
 
Lesson Takeaways
  • Know what the Power Rule says and how to apply it to polynomials terms
  • Be comfortable doing integer operations mentally, with or without coefficients
  • Be cautious and thorough with your work when taking the derivative of rational terms, radical terms, and terms with fraction exponents
  • Rewrite expressions in classic $x^n$ Power Rule form before you take the derivative, to not only show cleaner scratch work, but also avoid typical mistakes
  • Understand the difference between what can and cannot be done without needing more advanced rules, such as the upcoming Product and Chain rules

Lesson Metrics

At the top of the lesson, you can see the lesson title, as well as the DNA of Math path to see how it fits into the big picture. You can also see the description of what will be covered in the lesson, the specific objectives that the lesson will cover, and links to the section's practice problems (if available).

Key Lesson Sections

Headlines - Every lesson is subdivided into mini sections that help you go from "no clue" to "pro, dude". If you keep getting lost skimming the lesson, start from scratch, read through, and you'll be set straight super fast.

Examples - there is no better way to learn than by doing. Some examples are instructional, while others are elective (such examples have their solutions hidden).

Perils and Pitfalls - common mistakes to avoid.

Mister Math Makes it Mean - Here's where I show you how teachers like to pour salt in your exam wounds, when applicable. Certain concepts have common ways in which teachers seek to sink your ship, and I've seen it all!

Put It To The Test - Shows you examples of the most common ways in which the concept is tested. Just knowing the concept is a great first step, but understanding the variation in how a concept can be tested will help you maximize your grades!

Lesson Takeaways - A laundry list of things you should be able to do at lesson end. Make sure you have a lock on the whole list!

Special Notes

Definitions and Theorems: These are important rules that govern how a particular topic works. Some of the more important ones will be used again in future lessons, implicitly or explicitly.

Pro-Tip: Knowing these will make your life easier.

Remember! - Remember notes need to be in your head at the peril of losing points on tests.

You Should Know - Somewhat elective information that may give you a broader understanding.

Warning! - Something you should be careful about.

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