Derivatives of Sums and Constants

Lesson Features »

Lesson Priority: High

  • Learn how the derivative of a function is affected if the function is multiplied by a constant
  • Learn why the derivative of a constant on its own is zero
  • Understand why and how we can take the derivative of sums and differences by handling one term at a time
Lesson Description

Not coincidentally similar to limits, the derivative of a sum or difference of functions can be handled by taking things one piece at a time. This lesson outlines this for us with practice, and also looks at the derivative of a function that is multiplied by a constant.

Practice Problems

Practice problems and worksheet coming soon!


Derivative Sums and Differences

Derivatives are what we refer to as linear operators, and as such behave like objects do with the distributive property ».Most importantly, what this means for us in terms of actual usefulness is that derivatives of sums of terms will be equal to the sum of each term's derivative.
Example 1If the derivative of some object $a$ is $4x^3$ and the derivative of object $b$ is $\sqrt{x}$, what is the derivative of $(a+b)$?$\blacktriangleright$ The derivative of the sum of $a$ and $b$ is equal to the sum of each's derivative.$$\frac{d}{dx} \, (a+b) = \frac{d}{dx} \, a + \frac{d}{dx} \, b$$$$= 4x^3 + \sqrt{x}$$
It's almost as if the derivative operation acts like a variable, and that we literally applied the distributive property to the expression.For this reason, and also for the fact that addition and subtraction are essentially the same operation, this rule applies to differences as well as sums.
Derivatives of Sums and DifferencesLet $f(x)$ and $g(x)$ be differentiable functions. The derivative of the sum of $f$ and $g$ is equal to the sum of the derivative of $f$ and the derivative of $g$.Symbolically,$$\frac{d}{dx} \, \left[ f(x) \pm g(x) \right] = \frac{d}{dx} \, f(x) \pm \frac{d}{dx} \, g(x)$$
This rule is intuitive, and is the kind of action you will use so much that you won't even think of this as its own separate rule. One convenience that you'll quickly take for granted is polynomials, which are sums of terms by definition. It would be taxing to have to stop and work on a polynomial derivative » by separating it into separate terms first. Because of this rule, we will crunch polynomial derivatives as one step, as we will soon see.

Constant Multipliers

As you learn how to take derivatives, you'll see that it's more common than not to have a constant multiplier attached to the term that you're taking the derivative of. Fortunately, this common occurence will never change your approach, as constant multipliers simply "come along for the ride".
Derivatives of Terms with Constant CoefficientsLet $f(x)$ be a differentiable function, and let $f'(x)$ be its derivative. For some constant real number $c$, it follows that$$\frac{d}{dx} \, c\cdot f(x) = c \cdot f'(x)$$Stated differently, the derivative of a function $f(x)$ multiplied by $c$ is equal to $c$ times the derivative of $f(x)$.
For example, if you know the derivative of $\tan(x^2)$ is $2x \sec^2 (x^2)$, then the derivative of $5\tan(x^2)$ is $5 \cdot 2x \sec^2 (x^2)$ or $10x \sec^2 (x^2)$.I find most students take to this rule intuitively similar to the prior rule for sums and differences. However, take care to understand this conceptually. There is a common point of confusion when it comes to constants, because of the final rule of this lesson, and the best way to avoid confusing this rule with the next rule is to focus on why they are conceptually different.

Lone Constants

The derivative of a real number constant is zero. The foundation of this fact can be derived when we look at polynomial derivatives » but it is something we should know as a standalone rule.
Derivatives of Lone ConstantsLet $c$ be any real number. If $f(x) = c$, then$$\frac{d}{dx} f(x) = 0$$

Perils and Pitfalls

Take a moment to reflect on both of the constant rules in this lesson, and hopefully you can appreciate why it's possible to mix them up if you rush or if you aren't paying attention. Most importantly, make sure you can tell them apart and see them as two different rules for two different situations.
A super common mistake is to mix up the prior two constant rules. Try to understand them as what they are - two very different concepts. Constant multipliers are scale factors that change the size of the object to which they are attached, while lone constants are their own object.
Pro Tip
Because lone constants are a component of polynomials, the upcoming lesson on polynomial derivatives » will include rules that can help you remember why the derivative of a standalone constant is zero.

Mr. Math Makes It Mean

Learning these rules is important because they are the foundation of the mechanics of taking derivatives, and "taking derivatives" is the two-word summary of what your first few months of Calculus are focused on. However, because these are supporting rules and not independent rules, you won't be tested on this topic the same way you will be tested on future topics.Teachers like to use algebraic and graph based unknown functions to ask you whether or not you understand these rules. Many students find this wildly confusing at first. It's often something I categorize as mean not because it's difficult, but because it's often thrown at students with no explanation. Let's look at each.Graph ApproachUsing the provided graphs of $f(x)$ and $g(x)$, estimate the following derivatives.
Example 2$$2f'(1) - 3g'(6)$$$\blacktriangleright$ Since we do not have the function definitions, the best we can do is estimate. Sketch in your best tangent lines for each function at each point of interest, and estimate the slopes of those lines.I estimate $f'(1)$ to be about $1$, and $g'(6)$ to be about $1.5$. Therefore we have$$2f'(1) - 3g'(6)$$$$\approx 2(1) - 3(1.5) = -2.5$$
Example 3$$f'(g'(0))$$$\blacktriangleright$ Once again, we can only do out best to estimate the tangent lines. However, we need to work inside out here.It's pretty clear from the graph that $g'(0)$ is $0$, so now the question is, what is $f'(0)$? I estimate the tangent line of $f$ at $x=0$ to be about $-1$.
You Should Know
Teachers are usually a bit lenient with your artistry and slope estimation skills. To be safe, just write down what you did to get your estimate. For example, if you used two points on your tangent line sketch to do a slope calculation, show that work. Even just saying "I am estimating the slope to be -1" explicitly in a sentence is helpful.
Algebra ApproachWithout knowing the function itself, we can be told specific values of the function's derivative, and then use that information to answer questions. For example, if we know that$$\begin{align} h'(0) = -2 \,\,\,\,\,\,\,\, & j'(0) = 3 \\ h'(-3) = 2 \,\,\,\,\,\,\,\, & j'(2) = 4 \end{align} $$then we could be asked questions like$$\frac{d}{dx} \, h(x) + j(x) \, \rvert^{x=0} = ?$$$$\longrightarrow h'(0) + j'(0)$$$$ = -2 + 3 = 1$$ $$-5h'(-3) + 2j'(2) = ?$$$$\longrightarrow -5(2) + 2(4) = -2$$ $$h'(-j'(0)) = ?$$$$\longrightarrow h'(-(3)) = h'(-3) = 2$$It's always some type of substitution game - try to understand what you're being asked for, but most importantly, don't rush!

Put It To The Test

Example 4If the derivative of $f(x) = e^{2x}$ is $f'(x) = 2e^{2x}$, what is the derivative of $3f(x)$?
Show solution
$\blacktriangleright$ The constant multiplier rule tells us that the derivative of $3f(x)$ is $3\cdot f'(x)$.$$\frac{d}{dx} \, 3f(x) = 3\cdot \left[2e^{2x} \right]$$$$=6e^{2x}$$
Example 5$$\frac{d}{dx} \, \left[ \sin(x) - 2\csc(x) \right]$$Given that$$f(x) = \sin(x) \longrightarrow f'(x) = \cos(x)$$and$$g(x) = \csc(x) \longrightarrow g'(x) = -\csc(x) \cot(x)$$
Show solution
$\blacktriangleright$ Using the given function names, we are being asked for$$\frac{d}{dx} \, \left[ f(x) - 2g(x) \right]$$Using the rules for sums, differences, and constants, we know that this is equal to$$\frac{d}{dx} \, f(x) - 2 \cdot \left(\frac{d}{dx} \, g(x) \right)$$$$=\cos(x) - 2\left(-\csc(x) \cot(x) \right)$$$$=\cos(x) + 2\csc(x) \cot(x)$$
Example 6What is the derivative of $f(x) = 6$?
Show solution
$\blacktriangleright$ The derivative of a lone constant is zero.
Examples 7-9Given the following information:$$\begin{align} f'(3) = 5 \,\,\,\,\,\,\,\,& f'(-1) = 3 \\ f'(0) = 1 \,\,\,\,\,\,\,\,& f'(1) = 5 \\ g'(-4) = 0 \,\,\,\,\,\,\,\,& g'(4) = 2 \\ g'(3) = 1 \,\,\,\,\,\,\,\,& g'(7) = 7 \end{align} $$Determine the value of each of the following.
Example 7$$\frac{d}{dx} \, \left[ f(x) + g(x) \right] \,\,\,\, \mathrm{at} \,\,\,\, x=3$$
Show solution
$\blacktriangleright$ Using the given info, the derivative of $f$ at $x=3$ is $5$, and the derivative of $g$ at $x=3$ is $1$.$$\frac{d}{dx} \, \left[ f(x) + g(x) \right] \,\,\,\, \mathrm{at} \,\,\,\, x=3$$$$=f'(3) + g'(3)$$$$= 5 + 1 = 6$$
Example 8$$2f'(-1) - 3g'(4)$$
Show solution
$\blacktriangleright$ Constant multipliers stick around, so we will just plug in the values from the given information.$$2f'(-1) - 3g'(4)$$$$=2(3) - 3(2) = 0$$
Example 9$$f'(g'(3))$$
Show solution
$\blacktriangleright$ Evaluate this expression from the inside out. $g'(3) = 1$, so $f'(g'(3)) = f'(1)$, and $f'(1) = 5$.
Lesson Takeaways
  • Look at derivatives of sums and differences as a queue to take the derivative of each term
  • Understand that constant multipliers "stick around" and multiply into whatever derivative you are finding
  • Know that the derivative of a constant on its own is zero, and separate why this is different from derivatives with constant multipliers
  • Be able to answer questions about unknown functions, with either graphs or given information

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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