# Math Help - I do not understand Leibniz notation

1. ## I do not understand Leibniz notation

So I have started a 3rd calculus class and my prof uses primarily leibniz notation. (Ex. $\frac{dy}{dx}$) and I don't really understand how it all works.

Like I get that when you write dy/dx it's basically the difference between x and y as that difference gets arbitrarily small.

I also understand that in an integral, the dx stands for a tiny verticle sliver and the integral means we are adding up an infinite number of those slivers.

I get lost after that. For example, how does the chain rule use this notation? I've heard there are benefits to it in explaining the chain rule, but I don't quite follow it. An example/explanation would be great.

Finally I am given equations such as:
$\int_a^b \sqrt{(dx)^2 + (dy)^2} dx$
for the arclength of a function and:

$\int_{t_1}^{t_2} \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} dt$
for the arclenth of a function given using a parametric equation.

So what do these mean? In the first one, what would I replace dx and dy with. How about in the second one, what is (dx/dt)?

-Keilan

2. Originally Posted by Keilan
So I have started a 3rd calculus class and my prof uses primarily leibniz notation. (Ex. $\frac{dy}{dx}$) and I don't really understand how it all works.

Like I get that when you write dy/dx it's basically the difference between x and y as that difference gets arbitrarily small.

I also understand that in an integral, the dx stands for a tiny verticle sliver and the integral means we are adding up an infinite number of those slivers.

I get lost after that. For example, how does the chain rule use this notation? I've heard there are benefits to it in explaining the chain rule, but I don't quite follow it. An example/explanation would be great.

Finally I am given equations such as:
$\int_a^b \sqrt{(dx)^2 + (dy)^2} dx$
for the arclength of a function and:

$\int_{t_1}^{t_2} \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} dt$
for the arclenth of a function given using a parametric equation.

So what do these mean? In the first one, what would I replace dx and dy with. How about in the second one, what is (dx/dt)?

-Keilan
Just think of $dx$ and $dy$ as very small quantities, but still numbers. (Don't just treat them like symbols.) For the arc length one, you might see it as $\int_a^b\sqrt{(dx)^2+(dy)^2}$, but watch how it transforms to something more understandable.

$\int_a^b\sqrt{(dx)^2+(dy)^2}=\int_a^b\sqrt{(dx)^2\ left(1+\frac{(dy)^2}{(dx)^2}\right)}$ $=\int_a^b dx\sqrt{1+\left(\frac{dy}{dx}\right)^2} = \int_a^b\sqrt{1+[f'(x)]^2}\,dx$

which is much easier to work with. The original formula comes from the Pythagorean Theorem, actually. The idea is that if you zoom in really close on any curve, a tiny part of it will look like a straight line, whose distance is given by $\sqrt{(\Delta x)^2+(\Delta y)^2}$.

where the small change in $x$ is $\Delta x$ and the small change in $y$ is $\Delta y$. To find the total length of the curve, you just add up all those little parts:

$\sum_{1}^{n}\sqrt{(\Delta x)^2+(\Delta y)^2}$

Letting $n\to\infty$ changes this into an integral (recall Riemann sums).

$\int_a^b\sqrt{(dx)^2+(dy)^2}$

3. Alright, just some brief followup questions here.

For an integral where I have (dx) or (dy) on their own, am I required to transform them into $\frac{dy}{dx}$ in order to be able to use the formula? Or would there be some way to use those numbers without putting them into that familiar form?

As for this one:
$
$
$\int_{t_1}^{t_2} \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2}
$

What is
$\frac{dx}{dt}$?

Like say for example we were given
$x(t) = 2t^2 + 5t$.

Would I just replace
$\frac{dx}{dt}$ with $4t + 5$?

4. Yes.

If $x(t)=2t^2+5t$ and $y(t)=t^3-x+3$ (for example), then

$\frac{dx}{dt}=4t+5$ and $\frac{dy}{dt}=3t^2-1$

Therefore the integral $\int_{t_1}^{t_2} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}\,dt$ becomes:

$\int_{t_1}^{t_2}\sqrt{(4t+5)^2+(3x^2-1)^2}\,dt$

From there it's just a regular integral w.r.t $t$.