# Derivative of Inverse Functions (a.k.a, How to Create Your Own Table of Derivatives)

Quick question. Are you currently (or have been) a student of **differential calculus** (a.k.a., Calculus I)? If so, maybe you can guess what is on that (potentially-cryptic*)* graph above?

What. Inverse functions of course! Just look at that beautiful shape and how well they correspond with each other!

(OK. Bonus point: can you guess *which* functions they are?)

Yep. There is definitely more than meet the eyes. Granted, **inverse functions** are studied even before a typical calculus course, but their *roles* and *utilities* in the **development of calculus** only start to become increasingly apparent, after the discovery of a certain formula — which related the derivative of an inverse function to its original function. And guess what? *That* — is the topic that we’ll be delving into in greater details today.

So… hang on tight there, as you are about to travel to the land of *exotically-strange functions* in a few seconds. π

# A Primer on Inverse Functions

In theory, given a function defined on an interval , the role of is to map to . That is, plays the role of the **referrer**, with being the **target** of . Collectively, the set of all targets of under forms the set commonly known as — or the **image** of under (think of the **candle-and-wall analogy** in optics). Under this terminology, the function is said to map to .

Presumably, if another function can be constructed to map *each* target back to its referrer, then this new function would be considered the **inverse function** of — or for short.

Of course. Not all functions can have inverse. However, for a certain class of functions that are deemed **injective** (i.e., functions where every referrer points to a different target), every target will have a *unique* referrer. This means that it would then be possible to define a function — which maps each target in back to its original referrer in . Such a function would then be rightfully considered the inverse of .

In the context of functions involving real numbers (i.e.,** real-valued functions**), the domains are usually a union of **open** or **close interval**, and in the context of calculus, the functions are generally assumed to be **differentiable** (and hence **continuous**). As it happens, when these assumptions are combined together, the results are a series of *fundamental* and *increasingly powerful* theorems about **invertible functions**:

Given a function defined on an interval (possibly with a larger domain), if the function is *injective*Β on , then — when the domain is restricted to — has an inverse with domain .

Given a function defined on an interval (possibly with a larger domain), if the function is *continuous* and *injective *on , then is either **strictly increasing**, or **strictly decreasing** on .

Given a function defined on an interval (possibly with a larger domain), if is *continuous* and *injective* on , then — the inverse of as defined on — is continuous throughout as well. Moreover, if is strictly increasing (decreasing) on ,Β then is strictly increasing (resp., decreasing) on as well.

A classic example illustrating these theorems would be the function . For example, when we take the domain of to be (i.e., the set of all **real numbers**), the function would map both and to the same number, and is therefore *not* invertible. However, when the domain is *restricted* to the set of **non-negative numbers**, would now become *injective* everywhere, and hence is by extension *invertible*, with the inverse being the function .

Also, since is *continuous* and *injective* on the set of non-negative numbers, by Theorem 2 should be **monotone** (or as in this case — strictly increasing) on this domain, and since continuous injective functions produce continuous inverses, the inverse of — the square root function — should be continuous (and strictly increasing) on its own domain as well — which it is!

Note that in general, since the graph of is the set of points and the set of points , the two functions are — graphically speaking — **mirror reflection** of each other along the diagonal axis .

Alternatively, since is also the inverse of , one can also think of the **graph** of as the set of points , and the graph of the set of points . Hence, it is fair to say — in our language at least — that and are **correlates** of each other (from the perspective of ), much like the same way that and are correlates of each other (from the perspective of ).

# Derivative of Inverse Functions

As it stands, mathematicians have long noticed the relationship between a point in a function and *its correlate in the inverse function*. More specifically, it turns out that the **slopes** of tangent lines at these two points are exactly reciprocal of each other! To be sure, here’s a neat animation from Dr. Phan to prove our sanity:

In **real analysis** (i.e., theory of calculus), this geometrical intuition would then constitute the backbone of what came to be known as the **inverse function theorem** (on the derivative of a inverse function), which can be proved using the three aforementioned theorems above:

Given a function *injectively* defined on an interval (and hence defined on ), is **differentiable** at if the expression makes sense. That is, if the original function is differentiable at the **correlate **of , with a derivative that is* not *equal to .

In which case, the derivative of at exists and is equal to the said expression:

(1)

In English, this reads:

**reciprocal**of the derivative of the original function — at its correlate. Math Vault

**Leibniz’s notation**:

which, although not useful in terms of calculation, embodies the essence of the proof.

# Applications of Inverse Function Theorem

All right. So how do we apply this theorem in practice? Well, for that purpose, here we lay out *7 *examples — for your own pleasure. π

## Example 1 — Linear Functions

If we let the original function to be , for instance, then, after solving for and interchanging the with , we get that . In which case, it is clear that and .

So, this means that we just found out that the derivatives are reciprocal of each other — without appealing to any higher mathematical machinery.

But let’s say that we were to find using **Inverse Function Theorem**, then here is what we would have to do:

- Find the correlate of
- Calculate the derivative of the original function — at this correlate
- Take the reciprocal

And once that’s done, the number obtained would then be the derivative of the inverse function — at .

OK. Let’s see what happens if we follow through the steps:

- , so the correlate of is (i.e., is to as is to ).
- Taking the reciprocal, we get , so that — as expected from our result before..

Of course. In this case, **Inverse Function Theorem** is not really necessary, but it does illustrate the mechanics of calculating the derivative of an inverse function fairly well to get the momentum going.

## Example 2 — Square Root Function

Now that we know that **square** and **square root** functions are inverses of each other (when we restrict the domain of Β to the *non-negative numbers*), we should be able to calculate the derivative of the square root function with the help of its inverse. To illustrate, here’s how we can find the derivative of at :

- Correlate of ? Just . So is to as is to .
- Derivative of the original function at ? .
- Reciprocal? .

Therefore, — exactly as one would expect from the use of **power rule**.

So all is good in this case. However, don’t commit the *non-mathematician-like* mistake of indiscriminately invoking **Inverse Function Theorem**Β when the preconditions are not satisfied.

In particular, the derivative of does not exist at , because derivative of the original function at its correlate is , so that if you choose to take the reciprocal anyway, your derivative will blow up big time. Why? Because the square root function actually has a **vertical tangent** at !

However, when , , which illustrates that while the derivative of can be proved from definition, the derivative of can be proved — with a bit more style — using **Inverse Function Theorem**!

## Example 3 — General Root Functions

While and are inverses of each other alright (), it still needs to be recognized that there are really *two kinds* of root functions: one where the root is an **even** (natural) number, and one where the root is **odd**.

Indeed, when the root is *even*, and are inverses of each other *only* insofar as the domain of is restricted to **non-negative numbers**, but when the root is *odd*, these two functions are inverses of each other under the full domain of (i.e., ).

Using the same steps as before, we can find the derivatives of at for the appropriate domains:

- Correlate of : just , or .
- Derivative of the original function at the correlate:
- Reciprocal:

So for the **even** root functions, for , and for the **odd** root functions, the same is true for (why?). Either way, the derivative is left undefined at .

For example, for , and for all .

And as with before, the formula for the derivative of root functions is really just an special instance of the **power rule**. In fact, one way to think about it, is that this is how the **power rule for derivatives** came about — the derivative of proved from definitions, and the derivative of proved using **Inverse Function Theorem**.

## Example 4 — Logarithmic functions

Depending on which statements are adopted as definitions, the derivative of **logarithmic functions** either follows immediately — as a corollary from the definition of logarithm, or is proved using the derivative of its inverse — the **exponential function**.

Under the natural base , the exponential function on implicitly* defines* the logarithmic function on (i.e., the set of **positive numbers**). Using **Inverse Function Theorem**, the derivative of at any can be calculated as follows:

- Correlate of :
- Derivative of the original function at the correlate:
- Reciprocal:

Voila! for , as expected.

Wait…what about logarithmic functions of any *arbitrary* base (, )? Well, the **change of base theorem** for logarithms tells us that:

Therefore, for :

For example, , and .

So all is still good. Moving on!

## Example 5 — Inverse Trigonometric Function: Arcsine

For an angle , denotes the **y-coordinate** of a terminal point with angle . For example, , , and when is between and , is also squeezed between and .

In fact, it’s not too hard to visualize that as moves from to , increases strictly non-stop from to . This means that the function is *injective* on , and by extension, an inverse function of sine — denoted as or — can be defined on .

Naturally, this leads to a series of *calculus-related* questions:

- Where is this inverse sine function differentiable?
- What is the derivative in that case?

Luckily, this is where **Inverse Function Theorem** comes to the rescue big time:

- Correlate of : simply
- Derivative of the original function at the correlate:

OK. Before we move on, let’s simplify the expression a little bit. By **Pythagorean Theorem**, we know that , or that . In particular:

(e.g., if the argument of cosine ranges from to (i.e., stays on the *right* quadrants))

Fortunately, since for all by construction, the aforementioned expression becomes:

### Careful!
###### OK. That was pretty clean isn’t it? But that last equality need not hold if the domain of was restricted differently. For example, if the domain of were to be restricted to , would have mapped to , which means that would no longer apply. For this reason, is usually regarded as the **standard restricted domain** for sine — when it comes to defining the inverse of sine.

###### OK. That was pretty clean isn’t it? But that last equality need not hold if the domain of was restricted differently. For example, if the domain of were to be restricted to , would have mapped to , which means that would no longer apply. For this reason, is usually regarded as the **standard restricted domain** for sine — when it comes to defining the inverse of sine.

All right. That finishes Step 2, which means that when or , is not differentiable as would be . On the other side of the token, it also means that *would* be differentiable anywhere on the interval . Let’s finish up the third step in calculating the derivative of inverse sine then:

3. Reciprocal:

Now, putting everywhere together, we have that for . This means that while is defined and *continuous* on , it is only differentiable on the interval .

To be sure, here’s an accompanying graph of and — for the record:

## Example 6 — Inverse Trigonometric Function: Arccosine

For an angle , denotes the **x-coordinate** of a terminal point with angle . For example, , , and when is between and , is also squeezed between and .

Similar to the case with the **sine** function, it’s not too hard to visualize that as moves from to , decreases strictly from to non-stop. This means that the function is *injective* on , and by extension, an inverse function of cosine — denoted by or — can be defined on .

So, as constructed above, when is differentiable? And what would the derivative be in that case? Well, to find out more, let’s invoke **Inverse Function Theorem** again:

- Correlate of : just
- Derivative of the original function at the correlate:

And here’s the **fancy Pythagorean trick** again! First, , so that . In particular:

(e.g., if the argument of sine ranges from to (i.e., remains on the *upper* quadrants))

Luckily, since by construction ranges from to , the expression from Step 2 can be simplified as follows:

By inspection, this means that when is equal to or , is not differentiable as would be . However, the good news is that *would* be differentiable anywhere on . Finishing up Step 3, we get that:

3. Reciprocal:

which shows that for .

In other words, while arccosine is defined and *continuous* on , it is only differentiable on . Similiar to the arcsine function, arccosine also has vertical tangents at and .

And before we move on, let’s just mention that in general, the **standard restricted domain** for cosine is when it comes to defining inverse cosine, so that by convention, is typically understood to be *the* inverse cosine function mapping to . Were to be defined even slightly differently, the **fancy Pythagorean trick** introduced earlier could either *cease to work out properly* — or lead to a new, **non-standard formula** for the derivative of inverse cosine.

## Example 7 — Inverse Trigonometric Functions: Arctangent

GIven an angle , can be defined as the **ratio** between the y and x-coordinates of the terminal point with angle . Geometrically, in essence measures the **slope** between the origin, and the terminal point with angle .

For example, , because when the angle is , so does the slope. On the other hand, , because when the angle is degrees, the slope is .

In general, we interpret the **standard restricted domain** of the tangent function to be the interval . Under that interpretation, one can see that the *bigger* the angle, the *more positive* the slope. More specifically, as the angle moves from to , the tangent (i.e., slope) goes from *negatively sloppy*, *flat*, to *positively sloppy beyond bound*.

Algebraically, this means that the function is strictly increasing on as it travels from to , making it *injective* on . As a result, a inverse function of — usually denoted by or — can be defined on .

Hence by construction, has this *neat* property that it is defined everywhere and *continuous* on . Interesting! So how does behaves in terms of differentiability and derivatives then?

Well, there is one way we can find that out — through **Inverse Function Theorem** of course!

- Correlate of :
- Derivative of the original function at the correlate:

Now, to prevent us from *blindly manipulating symbols*, let’s just make sure that things still make sense here. We are talking about the derivative of , so could stand for any real number from to . By construction, Β (i.e., stays on the right side of the **right quadrants**). As a result, is hence well-defined everywhere on — regardless of the value of . Oh well, guess this means that everything is still on track!

Also, since we know that for , the aforementioned expression becomes:

which is really good, because for all . This means that the function is differentiable *everywhere* in its domain. Finishing Step 3 quickly yields:

Β Β Β Β Β Β Β 3. Reciprocal:

which means that for any . Perfection!

And to reward ourselves with some “math toys” for the great work being done, here’s a graph of both and — as defined using ‘s **standard restricted domain**:

# Closing Words and Beyond

So…is there more to this **Inverse Function Theorem** thingy? You bet! In particular, here are some recommended functions you might be tempted to differentiate — and add to your own list of favorite derivatives:

- (note: )

(heck, why not just construct your own **injectively differentiable** function from scratch, and find the derivative of its inverse?)

Either way, the take-home lesson — if there is any — is that by acknowledging that some functions come in pairs, and by learning how to *maneuver* around the formula for the derivative of an inverse, one can double the number of functions in their own **table of derivatives**, thereby doubling their knowledge on derivatives in general.

And before we go, here is a summary of what we have *invented*/*discovered* so far:

For all , .

OK. Enough of inverse functions for now. Tune in next time for more math goodies perhaps? In the meantime, you can drop by our Facebook and satisfy your craving for more **edutaining tidbits** and “**math cookies**“. π

Derek Pareto

March 3, 2016 @ 3:39 pm

Oh man. I wish they presented it the way you did when I learnt Cal 1. Excellent work!

Math Vault

March 3, 2016 @ 7:41 pm

Glad you enjoy it. Thanks!

Anitej Banerjee

May 18, 2016 @ 10:05 pm

This is amazing!

This site has been so helpful in making Calculus easier for me π (I hate having to remember the derivatives of arcsec and such functions, and I’m loving all the e and ln(x) derivations you guys do!)

Kudos to the writers! π

Math Vault

May 19, 2016 @ 12:30 am

Thank you! More derivative goodies coming up this month! π