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MindMap Gallery Concepts and calculations of differential calculus of functions of one variable

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.

Edited at 2022-07-03 07:56:52

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Concepts and calculations of differential calculus of functions of one variable

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Outline

Geometric Applications of Differential Calculus of Functions of One Variable
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Differential calculus of multivariate functions and its applications
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Differential calculus mind map
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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