Information on the use of the program
Understanding
Function studies on extremes, inflection points, monotony behavior, etc. are an important part of mathematics teaching in high school. The secure computing the derivative of a function is one of the essential basic techniques here. In the calculations, the derivation rules (sum and factor rule, product rule, chain rule and quotient rule) have a key role.
This program assumes that the derivatives important basic functions are known:
(X ^ n) '= n · x ^ (n-1) (sin x)' = x cos (cos x) '= - x sin (exp x)' = exp x
(Ln x) '= 1 / x
This refers as usual exp x = the exponential function x -> e ^ x with the Euler number e = 2.71828 .... Since the rule (x ^ n) '= n · x ^ (n-1) is valid for arbitrary real numbers, they detected, in particular the derivative of the root (x) = x ^ (1/2).
In the exercises, the input of operating rules to no specific form is attached. Whether, for example, the derivation of x · exp (2x) in the form 2x · exp (2x) + exp (2x) or (1 + 2x) exp (2x) is input, is irrelevant for the test, it is only u. U made. attention to simplification possibilities of calculation rule. Multiplication sign can where they can be omitted after the usual convention in mathematics, accounting.
The same applies to parentheses: sin x = sin (x), but of course is sin (2x) not sin 2x.
Unwanted or as yet unknown functions can also be clicked away in the exercises.
Basically, the program has three parts:
In the exercises 1-7 the safe handling of the derivation rules stated in each case is examined.
If this causes difficulties, helping the diagnosis exercises 8 and 9. They ensure joined by Aids with differentiated error messages that the differentiation rules are correctly applied.
Exercise 10 finally controls the correct calculation of the derivation of free input function rules.
The exercises provide essential learning tools that are not possible by the book literature. The tasks only form a small but significant excerpt from the complete range of school mathematics. If you are interested in an electronic program package for the whole school mathematics with all types of tasks and all the derivations, you can find it with the different license types (single license, education license with or without students copy license) under www.KLSoft.de.
Of course we also like to give more accurate information inquiries. The program package is used at several hundred schools.
The individual exercises:
Exercise 1 (sum and factor control):
The two rules are for differentiable functions like those mentioned above:
f (x) = g (x) + h (x) -> f '(x) = g' (x) + h '(x) f (x) = c · g (x) -> f' (x) = c · g '(x)
In the factor usually c denotes a constant number.
Example: f (x) = x + 3 * sin x gives f '(x) = 2x + 3 · cos
Exercise 2 (product rule):
For differentiable functions of this rule may be in the form
f (x) = g (x) · h (x) -> f '(x) = g' (x) · h (x) + g (x) * h '(x)
be specified.
Examples:
f (x) = sin x · x² has the derivative f '(x) = 2x · sin x + x² · cos
f (x) = sinx provides expx · f '(x) = sinx + expx expx · · · expx cosx = (sin x + cosx)
Exercise 3 and 4 (chain rule):
Probably the most important rule is the chain rule that leaves a note in the following form:
For f (x) = g (h (x)) is the derivative f '(x) = h' (x) · g '(y) where y = h (x), in brief f' (x) = h '(x) · g' (h (x))
Is called g (y) as the outer and h (x) as a function of the inner verkettenen function f (x) = g (h (x)).
Example:
For f (x) = sin (X) x -> h (x) = x² inner and y -> g (y) = siny external function. Because H '(x) = 2x and g' (y) = cos y with y = x² results in the derivative f '(x) = 2x · cos (X).
The program currently does not run on Intel CPUs.
You can download our free demo version of KLSoft3D for a functional test.
Understanding
Function studies on extremes, inflection points, monotony behavior, etc. are an important part of mathematics teaching in high school. The secure computing the derivative of a function is one of the essential basic techniques here. In the calculations, the derivation rules (sum and factor rule, product rule, chain rule and quotient rule) have a key role.
This program assumes that the derivatives important basic functions are known:
(X ^ n) '= n · x ^ (n-1) (sin x)' = x cos (cos x) '= - x sin (exp x)' = exp x
(Ln x) '= 1 / x
This refers as usual exp x = the exponential function x -> e ^ x with the Euler number e = 2.71828 .... Since the rule (x ^ n) '= n · x ^ (n-1) is valid for arbitrary real numbers, they detected, in particular the derivative of the root (x) = x ^ (1/2).
In the exercises, the input of operating rules to no specific form is attached. Whether, for example, the derivation of x · exp (2x) in the form 2x · exp (2x) + exp (2x) or (1 + 2x) exp (2x) is input, is irrelevant for the test, it is only u. U made. attention to simplification possibilities of calculation rule. Multiplication sign can where they can be omitted after the usual convention in mathematics, accounting.
The same applies to parentheses: sin x = sin (x), but of course is sin (2x) not sin 2x.
Unwanted or as yet unknown functions can also be clicked away in the exercises.
Basically, the program has three parts:
In the exercises 1-7 the safe handling of the derivation rules stated in each case is examined.
If this causes difficulties, helping the diagnosis exercises 8 and 9. They ensure joined by Aids with differentiated error messages that the differentiation rules are correctly applied.
Exercise 10 finally controls the correct calculation of the derivation of free input function rules.
The exercises provide essential learning tools that are not possible by the book literature. The tasks only form a small but significant excerpt from the complete range of school mathematics. If you are interested in an electronic program package for the whole school mathematics with all types of tasks and all the derivations, you can find it with the different license types (single license, education license with or without students copy license) under www.KLSoft.de.
Of course we also like to give more accurate information inquiries. The program package is used at several hundred schools.
The individual exercises:
Exercise 1 (sum and factor control):
The two rules are for differentiable functions like those mentioned above:
f (x) = g (x) + h (x) -> f '(x) = g' (x) + h '(x) f (x) = c · g (x) -> f' (x) = c · g '(x)
In the factor usually c denotes a constant number.
Example: f (x) = x + 3 * sin x gives f '(x) = 2x + 3 · cos
Exercise 2 (product rule):
For differentiable functions of this rule may be in the form
f (x) = g (x) · h (x) -> f '(x) = g' (x) · h (x) + g (x) * h '(x)
be specified.
Examples:
f (x) = sin x · x² has the derivative f '(x) = 2x · sin x + x² · cos
f (x) = sinx provides expx · f '(x) = sinx + expx expx · · · expx cosx = (sin x + cosx)
Exercise 3 and 4 (chain rule):
Probably the most important rule is the chain rule that leaves a note in the following form:
For f (x) = g (h (x)) is the derivative f '(x) = h' (x) · g '(y) where y = h (x), in brief f' (x) = h '(x) · g' (h (x))
Is called g (y) as the outer and h (x) as a function of the inner verkettenen function f (x) = g (h (x)).
Example:
For f (x) = sin (X) x -> h (x) = x² inner and y -> g (y) = siny external function. Because H '(x) = 2x and g' (y) = cos y with y = x² results in the derivative f '(x) = 2x · cos (X).
The program currently does not run on Intel CPUs.
You can download our free demo version of KLSoft3D for a functional test.
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