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Concepts and calculations of differential calculus of functions of one variable
This is a mind map about the concepts and calculations of differential calculus of functions of one variable. The main content includes concepts, calculations of derivatives and differentials, test questions, and 1,000 questions.
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Concepts and calculations of differential calculus of functions of one variable
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Concept and calculation of differential calculus of functions of one variable
concept
Quotes
instantaneous rate of change
dA/dB is called the instantaneous rate of change of A to B
Lemney value
slope of tangent line
The concept of derivative
Two expressions of important derivatives
Incremental
function difference form
4.1
Three ways to say derivatives are equivalent
y=f(x) is differentiable at point x0
The derivative of y=f(x) exists at point x0
f'(x0)=A (A is a finite number)
Necessary and sufficient conditions for function f(x) to be differentiable at x0
Both the left and right derivatives exist and are equal
f'-(x0)＝f' (x0)＝A
Derivative does not exist
Study the tangent problem of y=lxl at x=0
Sharp point, the derivative at the turning point does not exist
Study y = 1/3 power of x
Infinite derivatives. Derivative does not exist
Normal
Negative reciprocal of derivative
The concept of higher order derivatives
incremental generalization
Exam questions
Definition of derivative
It can be disassembled because it is said that it can be derived from the 4.1 fraction. It can be disassembled by taking it apart and taking a look. It is true that the limit exists after being disassembled.
4.2, first change the element, make it simpler, and then find the limit
Extreme operation in progress
Propose timely factors whose limit is not zero
It cannot be broken down because it only talks about question types under continuous 4.3 continuous conditions.
Create conditions without conditions
Divide one term by one term
4.11
Important Theorem 4.3
Assume that f(x) is continuous at x=x0 and satisfies that when x tends to x0, limf(x)/x-x0=A, then f(x0)=0, f'(x0)=A
Derive an even number of times and the parity remains unchanged. Lead an odd number of times, swap parity
4.6
Proof on the exchange of parity and evenness after derivation and the invariance of periodicity after derivation
4.4 and 4.5
The concept of differential calculus
△y＝A△x o(x)
A△x is also called the linear principal part, also called the differential of y
A△x＝dy＝f′(x)△x＝f′(x)dx
△x＝dx
A＝f′(x)
4.7 4.8
If it can be differentiable, it must be differentiable. If it can be led, it must be differentiable.
Differentiable judgment
The ultimate higher order ratio △x=0
Same as multivariate function
Calculation of Derivatives and Differentials
Arithmetic
Write the right side of the four arithmetic operations and push the left side
The same is true for the derivative of the quotient
Multiply more than 3 expressions
4.9
Multiply 100 terms and convert them into two terms
Derivatives of piecewise functions
Derivation using derivative definition at segmentation points
difficulty
Use the derivative formula to find the derivative at non-segmented points
Derivative of lnlxl=1/x
Derivative of lnlg(x)l=g′(x)/g(x)
Derivative of a to the x power
a to the power of x lna
Derivative of a to the u(x) power
u(x) power of a×lna×u′(x)
4.12
Derivative of composite functions
one journey at a time
differential form invariance
df port=f'(port)d port
Pay attention to the position of the derivation symbol
Pay attention to observation. It is not necessary to find the derivative and then add the value.
4.14
Inverse function derivation
Suppose y=f(x) is differentiable, and f′(x)≠0
Then f'(x) must be always positive or always negative
Suppose y=f(x) is continuous, and f'(x)≠0
Then f(x) must be always positive or always negative
first derivative of inverse function
The derivative of the inverse function = the reciprocal of the derivative of the original function
second derivative of inverse function
Y′′xx＝-X′′yy/(X′y)3
X′′yy＝-Y′′xx/(Y′x)3
Comprehensive examination questions with cash points
4.17 The inverse function gives the value of y, and it requires x to be brought in.
Be sure to pay attention to whether the value taken when deriving the derivative is x or y
Derivatives of parametric equations
First derivative of parametric equation
Find the second derivative of a parametric equation
The second-order derivative and the second-order derivative of the inverse function are not well grasped.
Implicit function derivation
y is a function of x
Derive directly from both sides
Logarithmic derivation
When multiplying, dividing, initiating, and exponentiating multiple items
Generally, take the logarithm first and then derive the derivative
After taking the logarithm, bring the exponent of the logarithm to the front.
When taking the logarithm, if the range is not specified, the absolute value needs to be added
Differentiation method of power exponential function
First convert it into an exponential function and then derive the derivative
To take the logarithm is to find the derivative on both sides, but to take the exponent is to find only one side
y=x raised to the power of x
Image, image method, derivative method
y=x raised to the power of 1/x
Image, image method, derivative method
higher order derivatives
Find the nth derivative of a raised to the xth power
a to the power of x × (lna) to the power of n
Use induction
8 n-order derivative formulas
The nth derivative of (xe to the power of x) = (x n)e to the power of x
Using higher order derivatives to find derivative formulas
Yang Hui triangle
Similar to binomial expansion
Use Taylor formula🐻
First write the Taylor formula or McLaughlin formula of y=f(x), and then compare the coefficients to obtain the n-order derivative of f (X0)
1. Any function that is differentiable of infinite order can be written as Taylor expansion and Maclaurin expansion.
2. The question gives a specific infinite-order differentiable function y=f(x), which can be expanded into a power series through a formula. p61
3. According to the uniqueness of the expansion formula, by comparing the n-th power coefficient of (x-x0) in 1.2, we can obtain the n-order derivative of f (X0)
4.27
5! =120
Exam questions
Important limits exist and ≠0
Mother is 0, fader is 0
4.1
The child is 0, and the mother is 0
Make up the limit. Molecules have physics. When you see the square root, you must remember that molecules have physics.
4.3 Taylor’s formula proves that it is differentiable
It's normal, but it can't be done
Prove that it is differentiable, continuous, and when the limit exists
Just find the strongest one and prove it directly.
Exercise 4.2 The absolute value can be regarded as bounded
g''(0) exists
g'(x) exists in a certain neighborhood of 0
g(x) is second-order differentiable at x=0. Exercise 4.4★★★
You cannot use Lupida to find the second derivative
Lópida can be used if the function can be defined in the decentered neighborhood
It only says that there is a second derivative at a point, and you cannot use Lupida for the second derivative. Because there is no guidance elsewhere
g′′(0) exists, it can be inferred that g′(x) exists in its decentralized field.
⭐❤️So use the derivative definition to find
❤️Or use the expansion of Taylor's formula with Peiano remainder to find
Derivative definition at segmentation points
Derivatives found directly at non-segmented points
Piecewise function, zero left and right are the same expression
No need to divide the discussion between left and right
Derivative of power function
The result needs to be the simplest
Find higher order derivatives
No rules, make difficulties easy
Decompose first
Lower the power again
Then use the higher order derivative formula
For example, the n-order derivative of (xe to the power of x)
(x n)e to the power of
Find dy/d(x2)
directly viewed as the form of division
For example, the derivative of a function such as y=arcsinx is to write the inverse function first. Then use the inverse function derivation rule to find it.
y＝1-x/1 x Simplified operation＝-1 2/1 x
For formulas containing more than e, first logarithm and then derivation 4.7
1000 questions
Taylor can only find higher-order derivatives at 0 point
Higher-order derivatives to the nth power of (x-a), the nth derivative = n!, and the rest = ni(x-a)
5. When seeking the high-order derivative of a piecewise function, the derivative is also directly derived at the piecewise point without defining the derivative.
After calculating the second-order derivative, we found arctan1/x in the formula, so we used the derivative definition and then derived the result.
6. Understanding function increments and differential drawings, the situation is different when the second derivative is greater than zero and less than zero.
12.13 For derivatives of complex functions, take the logarithm first. For the derivation of a logarithmic function, bring the power to the front and then perform the derivation.
15. For the expansion formula of this question, there is no need for a negative sign.
Higher order derivatives
Lemnitz
For finding the nth derivative of f(1)
Taylor formula
For finding the nth derivative of f(0)
Induction
For the nth derivative of f(x)
16. Find the value after changing yuan. Pay attention to whether the letters in the question are the same.
When using Lemnitz, whoever’s derivation becomes zero is written first.
Induction will also work
The reciprocal of 2/2 under the root = square root 2
On the second derivative formula of parametric equations
If the first derivative is complicated, you can use the formula to find the second derivative
Formula: y"t×x′t-y′t×x"t/(x′t)3
Differential dy = derivative × dx
24. Find the n-order derivative at f(0)
=g′(x) deduces that it is n 1th order
When deriving the derivative, n 1 needs to be multiplied forward.
g(x)=e to the power of x-1/x. It follows that the power of e to the power of x is expanded to n to the power of 2/x=n to the power of 1