Differential calculus is an important part of calculus, and its basic concepts are derivative and differential
This chapter covers the concept of derivatives, the rules for derivation of functions, higher-order derivatives, implicit functions and derivatives of functions determined by parametric equations, and differentiation of functions
1. Derivative concept
The basic concept of differential calculus - derivative
Definition of Derivatives: No more details
The limit of the ratio of the increment of the dependent variable and the increment of the independent variable exists, which is called the function can be derived at a certain point, otherwise, the function is called non-derivable at a certain point. If it is not derivable due to the limit being infinite, for the sake of convenience, we also call the derivative of the function at this point infinite, which is essentially non-derivable.
The definition domain of the power function is related to the choice of the exponent
The left and right limits of the ratio are called the left and right derivatives, and the left and right derivatives are collectively called the one-sided derivative
A necessary and sufficient condition for a function to be differentiable at a point is that the left and right derivatives of the function at that point both exist and are equal
Note the notation of left and right derivatives
A necessary and sufficient condition for a function to be differentiable at a point is that both the left and right derivatives exist and are equal
Functions are differentiable on a closed interval: continuous comparison with functions on a closed interval
Relationship between Tangent Slope and Normal Slope
A tangent at a point is not the same as a tangent through a point
If a function is differentiable at a certain point, then the function must be continuous at that point. A function is continuous at a point but not necessarily differentiable at that point. That is, the continuity of a function at a certain point is a necessary condition for the function to be derivative at that point, but it is not a sufficient condition
2. The derivation rule of the function
Use the derivation formulas of basic elementary functions and the basic derivation rules to differentiate elementary functions
The derivation formulas and basic derivation rules of basic elementary functions should be used proficiently
The derivation rule of sum, difference, product and quotient of functions
The laws of sum, difference and product can be extended to the case of any finite number of differentiable functions
The derivation rule of the inverse function: pay attention to the preconditions required by the rule
The derivative of the inverse function is equal to the reciprocal of the derivative of the direct function
Derivative rule of compound function
When applying the compound function derivation rule, the first step is to analyze which functions can be regarded as compounded by the given function.
The derivation rule of composite functions can be extended to the case of multiple intermediate variables
3. Higher order derivatives
Note the notation of higher derivatives
A function has an n-order derivative, and it is also often said that a function is n-order derivable. If a function has a derivative of order n at a point, then the function must have all derivatives below order n in some neighborhood of that point. Derivatives of the second order and above are collectively referred to as higher order derivatives
If you need to find the higher-order derivative formula of a function, you need to be good at finding some kind of law in the process of successive derivation
If the functions u and v both have n derivatives at point x, then obviously u+v and uv also have n derivatives at point x
Leibniz's formula: the n-order derivative formula of the product of two functions, which can use the binomial theorem to help memorize
4. Implicit functions and derivatives of functions determined by parametric equations
The left end of the equal sign is the symbol of the dependent variable, and the right end is the expression containing the independent variable. When the independent variable takes any value in the domain, this expression can determine the corresponding function value. Functions expressed in this way are called explicit functions
In general, if the variables x and y satisfy an equation F(x,y)=0, under certain conditions, when x takes any value in a certain interval, there is always a unique y value that satisfies the equation. , then it is said that the equation F(x,y)=0 determines an implicit function in this interval
The explicitness of implicit functions is sometimes difficult, if not impossible. However, in practical problems, sometimes it is necessary to calculate the derivative of the implicit function. Therefore, we hope to have a method that can directly calculate the derivative of the implicit function determined by the equation regardless of whether the implicit function is explicit or not.
If the implicit function is second-order derivable, then take one more step to obtain the second-order derivative of the implicit function.
The derivative of the power index function is generally calculated using the logarithmic derivation method: there are two ways p104
Derivative of a function determined by a parametric equation
Sometimes it is difficult to eliminate parameters
Use the formula directly (personally think it should meet the conditions of use)
Correlated rate of change problem: finding one rate of change for another rate of change
5. Differentiation of Functions
Definition of function differentiation: no more details
A necessary and sufficient condition for a function to be differentiable at a point is that the function is differentiable at that point
The formula will be remembered when you use it a lot
The differentiation of a function at any point in its domain is called the differentiation of a function. Differentiation of a function with respect to x and deltax
The increment deltax of the independent variable x is usually called the derivative of the independent variable
The quotient of the differential dy of a function and the differential dx of the independent variable is equal to the derivative of the function
The geometrical meaning of differentiation: local linearization of nonlinear functions (near the tangent point)
Differential formulas and differential algorithms of basic elementary functions
Combined memory with derivative formulas and algorithms for derivation
Differential Laws of Compound Functions: Differential Form Invariance
The role of differentiation in approximate calculation: the use of differential can replace some complex calculation formulas with simple approximate formulas
deltay、f（x0+deltax）、f（x）
Five approximate formulas for p118
In fact, this approximate substitution is very similar to the equivalent infinitesimal (not all)
If the number of square root is higher, it can better reflect the superiority of approximate calculation by differential
Error estimation: clarify the concept and do a few more questions