by

Ciro Santilli (@cirosantilli, 32)

Derivative

Chain rule

Here's an example of the chain rule. Suppose we want to calculate:

\frac{d e ^{2} x}{d x}

So we have:

f (x) = e^{x} g (x) = 2 x

and so:

f^{'} (x) = e^{x} g^{'} (x) = 2

Therefore the final result is:

f^{'} (g (x)) g^{'} (x) = e^{2 x} 2 = 2 e^{2 x}

Multivariable chain rule

Differentiable function

Smoothness

Infinitely differentiable function ( $C^{\infty}$ )

Bump function

Flat top bump function

math.stackexchange.com/questions/1786964/is-it-possible-to-construct-a-smooth-flat-top-bump-function

Maxima and minima

Given a function

f

:

from some space. For beginners the real numbers but more generally topological spaces should work in general
to the real numbers

we want to find the points

x

of the domain of

f

where the value of

f

is smaller (for minima, or larger for maxima) than all other points in some neighbourhood of

x

.

In the case of Functionals, this problem is treated under the theory of the calculus of variations.

Lifegard problem

pumphandle.consulting/2020/09/04/the-lifeguard-problem-solved/

Derivative test

Saddle point

Newton dot notation

Partial derivative

Partial derivative notation

Partial derivative symbol ( $\partial$ )

Nope, it is not a Greek letter, notably it is not a lowercase delta. It is just some random made up symbol that looks like a letter D. Which is of course derived from delta, which is why it is all so damn confusing.

I think the symbol is usually just read as "D" as in "d f d x" for

\frac{\partial F ( x , y , z )}{\partial x}

.

Partial label partial derivative notation ( $\partial_{x} F$ , $\partial_{y} F$ )

Partial index partial derivative notation ( $\partial_{0} F$ , $\partial_{1} F$ )

This notation is not so common in basic mathematics, but it is so incredibly convenient, especially with Einstein notation as shown at Section "Einstein notation for partial derivatives":

\partial_{0} F (x, y, z) = \frac{\partial F ( x , y , z )}{\partial x} \partial_{1} F (x, y, z) = \frac{\partial F ( x , y , z )}{\partial y} \partial_{2} F (x, y, z) = \frac{\partial F ( x , y , z )}{\partial x}

This notation is similar to partial label partial derivative notation, but it uses indices instead of labels such as

x

,

y

, etc.

Total derivative

The total derivative of a function assigns for every point of the domain a linear map with same domain, which is the best linear approximation to the function value around this point, i.e. the tangent plane.

E.g. in 1D:

T o t a l d e r i v a t i v e = D [f (x_{0})] (x) = f (x_{0}) + \frac{\partial f}{\partial x} (x_{0}) \times x

and in 2D:

D [f (x_{0}, y_{0})] (x, y) = f (x_{0}, y_{0}) + \frac{\partial f}{\partial x} (x_{0}, y_{0}) \times x + \frac{\partial f}{\partial y} (x_{0}, y_{0}) \times y

Directional derivative

 Ancestors

 Incoming links

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