Question

Question from 1/2nd lecture:

I have trouble understanding the notations of calculus, including Professor Jerison's notations.

Derivatives started out being expressed as f'(x) using Newtonian notation. However, Dr. Jerison transitions into a Leibniz notation (df/dx = dy/dx = d/dx f = d/dx y) fairly early on in the lecture.

Can someone explain how each of the notations are the equal and why Professor Jerison has adopted the Leibniz notation?

Also, I suspect there is some subtlety to the expression 'x(subscript)0' but does Professor Jerison use it in the sense to mean 'when x is zero?

Answer 1

The apostrophe notation assumes we know what the independent variable is.
The notation f(x) means f is a function of x, and the independent variable is x
f'(x) then means the same thing as df/dx (derivative of f(x) with respect to x)
You may sometimes see a dot over a function
\(\dot{f} \), and this is due to Newton. It is understood that f is a function of "t" (i.e. time)
and \(\dot{f} \) corresponds to df/dt (derivative of f with respect to t) . (This shows up in physics)

Answer 2

\( x_0\) typically means a fixed or constant value, as contrasted to the variable \( x \)

Answer 3

Often, rather than
f(x) = some function of x
you see
y = some function of x
This notation corresponds to applications where we think in terms of cartesian (x,y) coordinates.
For example
\[ y= x^2\]
here, \( \frac{dy}{dx} = 2x \) , and equates to the elementary notion of slope of a line.

Answer 4

You also "differentials"
derivative of the left side = derivative of right side
for example
\[ y= x^2\]
and writing the differentials
\[ dy = 2x \ dx \]
This notation is perhaps dicey. Ratios of "tiny" numbers, e.g. dy/dx are no longer "tiny"
but dy or dx by itself are more problematic (though I guess some math guru can explain how to interpret them)

Answer 5

There is also "implicit differentiation"
where we take the derivative of each variable with respect to some variable
for example, say we want the derivative with respect to "t" (and we assume the variables change with t):
\[ \frac{d}{dt}\left(y= x^2 \right)\\\frac{d}{dt}y = \frac{d}{dt}x^2\\
\frac{dy}{dt}= 2x \frac{d}{dt}x \\
\frac{dy}{dt}= 2x \frac{dx}{dt}
\]

Answer 6

you also see "simplified" notation, often when presenting "rules" for differentiation.
For example the product rule may be written as
d (uv) = u dv + v du

which is a bit cryptic. but it means
"to take the derivative of the product of two variables (with respect to some third variable)"
form the product of the first variable and the derivative of the second variable, and ditto for the 2nd variable
and add the two products.

Answer 7

as a silly example of the product rule, and showing the notation
\[ \frac{d}{dx}x^2=\frac{d}{dx}( x\cdot x) = x \frac{d}{dx} x + x \frac{d}{dx} \\
= x \frac{dx}{dx}+ x \frac{dx}{dx} \\=x+x\\=2x
\]