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Derivative, also known as WeChat service, is a mathematical concept abstracted from speed problem and tangent problem. 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 the time T be s = f (t), then the average speed of the car from time t0 to t 1 is [f (T 1)-f (t0)/t1-t0]. At this time, the speed of the car will not change greatly, and the average speed can better reflect the car from t0 to t. 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)].
Derivative is an important concept in calculus. Derivative is defined as the limit of the quotient between the increment of dependent variable and the increment of independent variable when the increment of 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.
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.
Method of derivation
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(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);
②(x^n)'=nx^(n- 1)(n∈q)；
③(sinx)' = cosx；
④(cosx)' =-sinx；
⑤(e^x)'=e^x；
⑥ (a x)' = a A Xin (ln is natural logarithm).
(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!
Derivative formula and its proof
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The following will list the derivatives of several basic functions and their derivation processes:
1.y=c(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.y=logax y'=logae/x
y=lnx y'= 1/x
5.y=sinx y'=cosx
6.y=cosx y'=-sinx
7.y = Tanks Y' =1/cos 2x
8.y=cotx y'=- 1/sin^2x
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,⊿y=c-c=0,lim⊿x→0⊿y/⊿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，
⊿y=a^(x+⊿x)-a^x=a^x(a^⊿x- 1)
⊿y/⊿x=a^x(a^⊿x- 1)/⊿x
If you do ⊿x→0 directly, the derivative function cannot be derived, and an auxiliary function β = a ⊿ x- 1 must be set to substitute for the calculation. We can know from the auxiliary function: ⊿x=loga( 1+β).
So (a ⊿ x-1)/⊿ x = β/loga (1+β) =1/loga (1+β)1/β.
Obviously, when ⊿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 =
Substituting this result into lim⊿x→0⊿y/ ⊿ x = lim ⊿ x → 0ax (a ⊿ x-1)/⊿ x gives lim ⊿ x → 0 ⊿ y/.
We can know that when a=e, there is y = e x y' = e x
4.y=logax
⊿y=loga(x+⊿x)-logax=loga(x+⊿x)/x=loga[( 1+⊿x/x)^x]/x
⊿y/⊿x=loga[( 1+⊿x/x)^(x/⊿x)]/x
Because ⊿x→0, ⊿x/x tends to 0 and x/⊿x tends to infinity, lim ⊿ x→ 0 loga (1+⊿ x/x) = logae.
lim⊿x→0⊿y/⊿x=logae/x。
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
⊿y=sin(x+⊿x)-sinx=2cos(x+⊿x/2)sin(⊿x/2)
⊿y/⊿x=2cos(x+⊿x/2)sin(⊿x/2)/⊿x=cos(x+⊿x/2)sin(⊿x/2)/(⊿x/2)
So lim⊿x→0⊿y/⊿x=lim⊿x→0cos(x+⊿x/2)? lim⊿x→0sin(⊿x/2)/(⊿x/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.