I know nothing about differential equations

[mather]

One reason we study Differential equations is that in the world around us it is easy to observe and measure changes of quantities in time or space. Changes in temperature, changes in speed, changes in concentration, etc. We have found that if we can express those changes mathematically as differentials. This leads us to an equation containing expressions of differentials of our variables of interest. If we can solve these differential equations we end up with a function of that variable in time or space.

thanks!

as for my question above?

 

[mather]

hello!

can anyone add to this excellent animation, the animated graph of the second, third, etc derivative?

thanks!

anyone ?

 

[Mark44]

okay
$x'(t) = \frac{dx}{dt} = \frac{\partial x}{\partial t} = \frac{d}{dt} x(t) = \frac{\partial}{\partial t} x(t) = \dot{x}(t)$.

These deserve some comment. Assuming x is a function of a single variable t, then all of the above are different ways of writing the derivative of x with respect to t.

x'(t) and $\dot{x}(t)$ are variations of Newton's notation. For Newton, derivatives were always time derivatives; i.e., derivatives with respect to t. Newton used the dot notation, and the "prime" notation, as in x', is very similar.

The "d/dt" notation is due to Liebniz, who developed calculus at about the same time as Newton.

The notation with the "curly" d indicates that we're dealing with a partial derivative. That is, the function being differentiated has two or more variables, and we're looking at the (partial) derivative with respect to one of those variables.

If x is a function of only one variable, say t, then the partial (or partial derivative) of x with respect to t is exactly the same as the derivative of x with respect to to. OTOH, if x happens to be a function of, say, t and v, then the ordinary derivative is not defined, but the two partials are.

In other words, this is meaningless for a function of two or more variables: $$ \frac{dx}{dt}$$
but these have meaning: $$ \frac{\partial x}{\partial t} \text{and} \frac{\partial x}{\partial v}$$

Aummarizing it is the derivation of a function $x$ after $t$

It is the derivative of x with respect to t. Derivation has a different meaning.

 

[K41]

OP, is this what you are looking for:
http://www.math.umn.edu/~rogness/multivar/partialderivs.shtml