1 ) Theorem on the Limits of Rational Expressions
WHERE:
n = Rational number
a = Constant
NOTE: The limit of an isolated constant is the same constant. When a constant is multiplied by a function, that constant can be taken out of the limit. Limits can be applied to separate functions when adding, subtracting, multiplying, or dividing functions. When there are limits to the power of a function, that limit can be applied within the power.
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2 ) Theorem on the Limits of Trigonometric Expressions
NOTE: The limit here is close to zero. The contradiction of the limit theorem of trigonometric functions is 1.
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3 ) Differentiation notation
NOTE: When x is changed, the amount of y change is Differentiation. Differential is used in many practical situations.
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4 ) Definition of differentiation
WHERE:
Y = F( x )
NOTE : The differential definition is constructed using the limit. Differential definition equation is used to differentiate by the fundamental principles.
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Differentiation of polynomial rational functions
5 ) Standard differential formula
WHERE:
n = Rational number
NOTE: According to this equation, the differential of a constant is zero..
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6 ) Differentiation of (Power = 1st) function
WHERE:
Y or f( x ) = x
NOTE: When differentiating a power of 1 variable, the answer is 1. All functions can be defined as Y or F (x).
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7 ) Differentiation of a square root variable
WHERE:
Y = root x
NOTE: All functions can be defined as Y or F (x).
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8 ) Differentiation of a 1 divided by variable
WHERE:
Y = 1 / x
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Differential formulas of standard trigonometric functions
9 ) Differential of Sin( x )
WHERE:
Y = sin ( x )
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10 ) Differential of Cos( x )
WHERE:
Y = Cos ( x )
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11 ) Differential of tan( x )
WHERE:
Y = tan ( x )
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12 ) Differential of cot ( x )
WHERE:
Y = cot ( x )
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13 ) Differential of sec ( x )
WHERE:
Y = sec ( x )
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14 ) Differential of cosec ( x )
WHERE:
Y = cosec ( x )
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Differentiation of the standard exponential function
15 ) Differential of the exponential function
WHERE:
Y = e^x
NOTE: When an exponential function is differentiated, it returns the same function.
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Differentiation of the standard logarithmic function
16 ) Differentiation of a logarithmic function
WHERE:
Y = ln ( x )
Or
Y = log e ( x )
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17 ) Multiplication differential of two variables
WHERE:
U and V = There are two functions of x
NOTE: Here multiply the first function by the differential of the second function, then multiply the second function by the differential of the first function and then add.
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18 ) Division differential of two variables
WHERE:
U and V = There are two functions of x
NOTE: The first differential function here is U. Here the differential multiplication function performs a subtraction. Eventually the whole function is divided by V squared.
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19 ) Contradiction theory of differentiation
NOTE: This theory is used when an x function is given by y to differentiate. It is also used to prove the differential formulas of standard inverse functions.
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20 ) Theory of a function of a function
NOTE: This theory is used when y is a function of u and u is a function of x. simply put, this theory is used to differentiate one function from another.
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Differentiation of standard inverse trigonometric functions
21 ) Differential of Sin inverse ( x )
WHERE:
Y = Sin inverse ( x )
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22 ) Differential of Cos inverse ( x )
WHERE:
Y = Cos inverse ( x )
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23 ) Differential of Tan inverse ( x )
WHERE:
Y = Tan inverse ( x )
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24 ) Differential of Cot inverse ( x )
WHERE:
Y = Cot inverse ( x )
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25 ) Differential of Sec inverse ( x )
WHERE:
Y = Sec inverse ( x )
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26 ) Differential of Cosec inverse ( x )
WHERE:
Y = Cosec inverse ( x )
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27 ) High order differential coefficients
WHERE:
Y’ = Y's first differential
Y’’ = Second differentiation of Y
Y’’’ = Third differential of Y
NOTE: High order differentials are often used for practical purposes.
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Practical stages of Differentiation
28 ) Sequence of tangent or curve
WHERE :
m = Sequence of tangent or curve
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29 ) Sequence of a perpendicular
WHERE :
m’ = Sequence of perpendicular
NOTE: Further, the differential can be used to find the fixed point of a curve and the velocity, acceleration, etc.
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