Derivadas "equações do tipo infinito"

Derivatives equal to infinity!

First of all, recall: what is the derivative?

You, the excellent calculus learner you are, would say "Oh, that's just the slope of the tangent line at a point!"

And you are correct! That is the derivative...

You, the excellent calculus student, might be asking, "Mr. Teacher, what happens if the tangent line is vertical?"

That's a great question, You! And how convenient, I have an example of this:

That, my friends, is the graph of $y = \sqrt[3]{x}$, a perfect example of the infinite derivative!

Look at the point where $x=0$. Draw the vertical line--it's vertical! So yes, this situation could definitely happen, and in fact, it happens to other function as well, such as $y = \tan^{-1}{x}$.

But you may inquire: "What is the value of the derivative at that point?" The simple answer is: the derivative, $f'(x)$, does not exist! Just like how diving by zero is undefined, the derivative at that point is undefined!

This makes sense; the derivative expression for $y = \sqrt[3]{x}$ is $$f'(x) = \dfrac{1}{3\sqrt[3]{x^2}}$$

(If you don't know how I got that, don't worry right now; that's covered in other sections.) We can see that if we let $x=0$, the fraction collapses to $\frac{1}{0}$, which is undefined.

So that's about it for derivatives equal to infinity!

When are derivatives equal to infinity? Let's take an example.

The reason is that $\sqrt{x}=x^{0.5}$. The derivative of $x^n$ is $nx^{n-1}$, or $0.5x^{-0.5}$.

Now what if we wanted the derivative of $\sqrt{x}$ at $x=0$? We plug $x=0$ into

$\frac{1}{2\sqrt{x}}$. THe denominator is 0!!!

That means that the limit approaches infinity.

In general, the derivative of $x^n$ at $x=0$ is infinite if $n$ is less than $1$, because

$(x^n)' = nx^{n-1}$, and when the exponent is negative, the x goes into the denominator.

We know that the derivative of $f$ evaluated at $a$ is defined as a limit. We know that limits can "equal" infinity.

Therefore there is a possibility that derivatives can "equal" infinity. In fact, this happens quite often when we're dealing with rational functions. All it means is that the graph goes essentially vertical at that point.