Derivative derivative is an important basic concept in calculus. The limit of the quotient between the increment of the dependent variable and the increment of the independent variable when the increment of the independent variable tends to zero. When a function has a derivative, it is said to be derivative or differentiable. The differentiable function must be continuous. Discontinuous functions must be non-differentiable. Derivative is essentially a process of finding the limit, and the four algorithms of derivative come from the four algorithms of limit.

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Derivative function

Derivative is an important concept in calculus.

Method of derivation

Derivative formula and its application in derivation

Solution of higher derivative

Derivative function

Derivative is an important concept in calculus.

Method of derivation

Derivative formula and its application in derivation

Solution of higher derivative

Derivative function

Also known as Ji Shu and WeChat services, they are mathematical concepts abstracted from the problems of speed change and curve tangent. Also known as the rate of change.

For example, if a car walks 600 kilometers per hour/kloc-0, its average speed is 60 kilometers per hour, but in the actual driving process, the speed changes, not all of them are 60 kilometers per hour. In order to better reflect the speed change of the car during driving, the time interval can be shortened. Let the relationship between the position S of the car and time t be S = f (t), then the average speed of the car during the period from time t0 to t 1 is [f (t1)-f (t0)]/[t1-t0]. The speed change of the car will not be great, and the average speed can better reflect the movement change of the car from t0 to t 1. Naturally, the limit [f (t1)-f (t0)]/[t1-t0] is taken as the instantaneous speed of the car at t0, which is also commonly known as the speed. Generally speaking, if the unary function y = f (x) is defined near the point x0 (x0-a, x0+a), when the increment of the independent variable δ x = x-x0 → 0, the limit of the ratio of the function increment δ y = f (x)-f (x0) to the increment of the independent variable exists and is limited, so the function f is derivable at the point x0. If the function f is differentiable at every point in the interval I, a new function with I as the domain is obtained, which is called f', called the derivative function of f, or derivative for short. The geometric meaning of the derivative f'(x0) of the function y = f (x) at x0: it represents the tangent slope of the curve l at P0 [x0, f (x0)]. Generally speaking, we get the law of judging the increase or decrease of function by derivative: let y = f (x) be derivable in (a, b). If in (a, b) and f' (x) >: 0, then f(x) increases monotonically in this interval. . If in (a, b), f' (x) < 0, then f(x) decreases monotonically in this interval. So when f'(x)=0, y = f (x) has a maximum or minimum, the maximum is the maximum and the minimum is the minimum.

The geometric meaning of derivative is the tangent slope of function curve at this point.

[Edit this paragraph] Derivative is an important concept in calculus.

Another definition of derivative: when x=x0, f' (x0) is a definite number. In this way, when x changes, f'(x) is a function of x, which we call the derivative function of f(x).

The derivative of y=f(x) is sometimes recorded as y', that is, f' (x) = y' = lim δ x→ 0 [f (x+δ x)-f (x)]/δ x.

Some important concepts in physics, geometry, economics and other disciplines can be expressed by derivatives. For example, derivatives can represent the instantaneous speed and acceleration of a moving object, the slope of a curve at a certain point, and the margin and elasticity in economics.

The definition of classical derivative mentioned above can be considered as a function change reflecting local Euclidean space. In order to study the change of vector bundle cross section (such as tangent vector field) on a more general manifold, the concept of derivative is extended to the so-called "connection". With connection, people can study a wide range of geometric problems, which is one of the most important basic concepts in differential geometry and physics.

Note: 1.f' (x) < 0 is a necessary and sufficient condition for f(x) to be a subtraction function, but it is not a necessary and sufficient condition.

2. The point where the derivative is zero is not necessarily the extreme point. When the function is a constant function, there is no increase or decrease, that is, there is no extreme point. But the derivative is zero. (The point with zero derivative is called stagnation point. If the signs of the derivatives on both sides of the stagnation point are opposite, it is an extreme point, otherwise it is a general stagnation point, such as f' (0) = 0 in y = x 3 and the sign of the derivative around x=0 is positive, it is a general stagnation point. )

[Edit this paragraph] Method of Derivation

(1) Find the derivative of the function y=f(x) at x0:

① Find the increment δ y = f (x0+δ x)-f (x0) of the function.

② Find the average change rate.

③ Seek the limit and derivative.

(2) Derivative formulas of several common functions:

① C'=0(C is a constant function);

②(x^n)'= nx^(n- 1)(n∈q)；

③(sinx)' = cosx；

④(cosx)' =-sinx；

⑤(e^x)' = e^x；

⑥ (a x)' = a xlna (ln is natural logarithm)

⑦ (Inx)' = 1/x(ln is natural logarithm)

⑧(logax)‘=(xlna)^(- 1),(a>； 0 and a is not equal to 1)

Add something. The above formula can not replace constants, but only functions. People who are new to derivatives often ignore this point, which leads to ambiguity. We should pay more attention to it.

(3) Four algorithms of derivative:

①(u v)'=u' v '

②(uv)'=u'v+uv '

③(u/v)'=(u'v-uv')/ v^2

(4) Derivative of composite function

The derivative of the compound function to the independent variable is equal to the derivative of the known function to the intermediate variable, multiplied by the derivative of the intermediate variable to the independent variable-called the chain rule.

Derivative is an important pillar of calculus. Newton and Leibniz made outstanding contributions to this!

[Edit this paragraph] Derivative formula and its proof

The following will list the derivatives of several basic functions and their derivation processes:

The basic derivative formula of1.y = c (where c is a constant) y'=0

2.y=x^n y'=nx^(n- 1)

3.y=a^x y'=a^xlna

y=e^x y'=e^x

4 . f(x)= logaX f '(x)= 1/xlna(a & gt； 0 and a are not equal to 1, x >；; 0)

y=lnx y'= 1/x

5.y=sinx y'=cosx

6.y=cosx y'=-sinx

7.y = Tanks Y' =1/(cosx) 2

8.y=cotx y'=- 1/(sinx)^2

9 . y = arcsinx y'= 1/√ 1-x^2

10 . y = arc cosx y'=- 1/√ 1-x^2

1 1 . y = arctanx y'= 1/( 1+x^2)

12 . y = arccotx y'=- 1/( 1+x^2)

In the process of derivation, there are several commonly used formulas to be used:

1.y=f[g(x)]，y'=f'[g(x)]？ G'(x) in' f' [g(x)], g(x) is regarded as a whole variable, while in G' (x), x is regarded as a variable. "

2.y=u/v,y'=(u'v-uv')/v^2

3. If the inverse function of y = f (x) is x=g(y), then y'= 1/x'

Certificate: 1. Obviously, y=c is a straight line parallel to the X axis, so the tangents everywhere are parallel to X, so the slope is 0. The definition of derivative is the same: y = c, δ y = c-c = 0, lim δ x→ 0 δ y/δ x = 0.

2. The derivation of this is not proved for the time being, because if it is deduced according to the definition of derivative, it cannot be extended to the general case that n is an arbitrary real number. Two results, y = e x y' = e x and y=lnx y'= 1/x, can be proved by the derivative of composite function.

3.y=a^x，

δy=a^(x+δx)-a^x=a^x(a^δx- 1)

δy/δx=a^x(a^δx- 1)/δx

If you do Δ x→ 0 directly, the derivative function cannot be derived, and an auxiliary function β = a Δ x-1must be set to substitute for the calculation. From the auxiliary function, we can know: Δ x = loga (1+β).

So (a δ x-1)/δ x = β/loga (1+β) =1/loga (1+β)1/β.

Obviously, when δ x→ 0, β also tends to 0. And lim β→ 0 (1+β) 1/β = e, so lim β→ 01/loga (1+β)1/logae =

Substitute this result into lim δ x → 0δ y/δ x = lim δ x → 0ax (aδ x-1)/δ x to get lim δ x → 0δ y/δ x = a xlna.

We can know that when a=e, there is y = e x y' = e x

4.y=logax

δy=loga(x+δx)-logax=loga(x+δx)/x=loga[( 1+δx/x)^x]/x

δy/δx=loga[( 1+δx/x)^(x/δx)]/x

Because when Δ x→ 0, Δ x/x tends to 0 and x/Δ x tends to ∞, so lim Δ x→ 0 loga (1+Δ x/x) = logae, so there is

limδx→0δy/δx = logae/x .

You can further use the bottoming formula.

limδx→0δy/δx=logae/x=lne/(x*lna)= 1/(x*lna)=(x*lna)^(- 1)

It can be known that when a=e, there is y = lnxy' =1/x.

At this point, y = x n y' = NX (n- 1) can be deduced. Because y = x n, y = e ln (x n) = e nlnx,

So you're NLNX? (nlnx)'=x^n？ n/x=nx^(n- 1)。

5.y=sinx

δy = sin(x+δx)-sinx = 2cos(x+δx/2)sin(δx/2)

δy/δx = 2cos(x+δx/2)sin(δx/2)/δx = cos(x+δx/2)sin(δx/2)/(δx/2)

So lim δ x→ 0 δ y/δ x = lim δ x→ 0 cos (x+δ x/2)? limδx→0 sin(δx/2)/(δx/2)= cosx

6. Similarly, y=cosx y'=-sinx can be deduced.

7.y=tanx=sinx/cosx

y'=[(sinx)'cosx-sinx(cosx)']/cos^2x=(cos^2x+sin^2x)/cos^2x= 1/cos^2x

8.y=cotx=cosx/sinx

y'=[(cosx)'sinx-cosx(sinx)']/sin^2x=- 1/sin^2x

9.y=arcsinx

x=siny

X' = comfort

y'= 1/x'= 1/cosy= 1/√ 1-sin^2y= 1/√ 1-x^2

10.y=arccosx

X = comfort

x'=-siny

y'= 1/x'=- 1/siny=- 1/√ 1-cos^2y=- 1/√ 1-x^2

1 1.y=arctanx

x=tany

x'= 1/cos^2y

y'= 1/x'=cos^2y= 1/sec^2y= 1/ 1+tan^2x= 1/ 1+x^2

12.y=arccotx

x=coty

x'=- 1/sin^2y

y'= 1/x'=-sin^2y=- 1/csc^2y=- 1/ 1+cot^2y=- 1/ 1+x^2

In addition, in the derivation of complex compound functions such as hyperbolic functions shx, chx, thx, inverse hyperbolic function ARSHX, ARCX, ARTHUX, etc., by consulting the derivation table and using the initial formula and

4.y=u soil v, y'=u soil v'

5.y=uv，y=u'v+uv '

You can get results quickly.

For y = x n y' = NX (n- 1) and y = a x y' = a xlna, there is a more direct derivation method.

y=x^n

According to the definition of exponential function, y>0

Take the natural logarithm of both sides of the equation.

ln y=n*ln x

Derive x on both sides of the equation. Note that y is a composite function of y and X.

y' * ( 1/y)=n*( 1/x)

y ' = n * y/x = n * x^n/x = n * x ^(n- 1)

Power function can also be proved in the same way.

To put it bluntly, the derivative is actually the slope.

Of course, the denominator mentioned above tends to zero, but don't forget that the numerator may also tend to zero, so the ratio of the two may be a certain number. If molecules tend to a certain number instead of zero, then the ratio will be very large, which can be considered infinite, that is, what we call derivative does not exist.

X/x, if x tends to zero here, the denominator tends to zero, but their ratio is 1, so the limit is 1.

It is recommended to know what the limit is first. Limit is an unattainable concept. You can get close to it, but you will never get there.

And realize that the derivative is a ratio.

Application of derivative

Monotonicity of 1. function

(1) Use the sign of the derivative to judge the increase or decrease of the function.

Using the sign of derivative to judge the increase or decrease of function is an application of the geometric meaning of derivative in studying the law of curve change, which fully embodies the idea of combining numbers with shapes.

Generally speaking, in a certain interval (a, b), if f' (x) > 0, then the function y=f(x) monotonically increases in this interval; If f' (x) < 0, the function y=f(x) monotonically decreases in this interval.

If there is always f'(x)=0 in an interval, then f(x) is a constant function.

Note: in a certain interval, F' (x) > 0 is a sufficient condition for f(x) to be increasing function in this interval, but it is not a necessary condition. For example, f(x)=x3 is the increasing function in R, but f'(x)=0 when x=0.

(2) the step of finding the monotone interval of the function

① Determine the domain of f(x);

② Deduction;

(3) Use (or) to solve the range corresponding to x. When f' (x) > 0, f(x) is increasing function in the corresponding interval; When f' (x) < 0, f(x) is a decreasing function in the corresponding interval.

2. Extreme value of function

Determination of Extreme Value of (1) Function

(1) If both sides have the same sign, it is not the extreme point of f(x);

(2) if it is near the left and right, it is the maximum or minimum.

3. Steps to find the extreme value of the function

(1) Determine the functional domain;

② Deduction;

③ Find all the stationary points in the definition domain, that is, find all the real roots of the equation;

(4) Check the symbols around the stagnation point. If Zuo Zheng is negative to the right, then f(x) gets the maximum at this root; If the left is negative and the right is positive, then f(x) takes the minimum value at this root.

4. The maximum value of the function

(1) If the maximum (or minimum) of f(x) in [a, b] is obtained at a point in (a, b), it is obvious that this maximum (or minimum) is also a maximum (or minimum), which is all the maximum of f(x) in (a, b).

(2) the step of finding the maximum and minimum value of f(x) on [a, b]

① Find the extreme value of f(x) in (a, b);

② Compare f(x) with the extreme values of f(a) and f(b), where the largest is the maximum value and the smallest is the minimum value.

5. Life optimization

In life, we often encounter problems such as maximizing profits, saving materials and achieving the highest efficiency. These problems are called optimization problems, and optimization problems are also called maximum problems. Solving these problems has important practical significance. These problems can usually be transformed into function problems in mathematics, and then into the problem of finding the maximum (minimum) value of the function.