Here's an example of the chain rule. Suppose we want to calculate:So we have:and so:Therefore the final result is:
math.stackexchange.com/questions/1786964/is-it-possible-to-construct-a-smooth-flat-top-bump-function
Given a function :we want to find the points of the domain of where the value of is smaller (for minima, or larger for maxima) than all other points in some neighbourhood of .
- from some space. For beginners the real numbers but more generally topological spaces should work in general
- to the real numbers
In the case of Functionals, this problem is treated under the theory of the calculus of variations.
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.
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":
This notation is similar to partial label partial derivative notation, but it uses indices instead of labels such as , , etc.
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.
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This is a section about Derivative!
For example, this sentence. And then another one.
The derivative measures the instantaneous rate at which a function changes. For a real function defined near , it exists when the difference quotient has a finite two-sided limit:The quotient is the slope of the straight line joining two nearby points of the graph. Its limit is the slope of the tangent line. Thus a derivative describes a local rate without choosing a particular finite measurement interval.
For , expanding the numerator givesAt , the increments , and give secant line slopes , and . The limit is , and the tangent line is . The picture makes the limiting process visible.
The same calculation shows why the derivative is useful for approximation:This is a local linear approximation. Near , replacing by makes the exact error , smaller than the leading change when is small. A curved graph can therefore be studied locally using a straight line.
If the function records position against time, its derivative is velocity. If it records an objective function, its derivative tests local changes that matter in optimization; in several variables, the gradient collects these directional rates and guides gradient descent. The interpretation comes from the function and its units, rather than from a new rule for each application.
Existence matters. For the absolute value function , the one-sided derivatives at zero are and , so there is no single tangent line described by a finite derivative there. Nor does a zero derivative alone identify a maximum or minimum: has but keeps increasing through zero.
This is a section about Derivative!
For example, this sentence. And then another one.
