Calculus derivative proof Let f (x) be differentiable at point x0, α n, β N is a positive sequence tending to zero, and it is proved that LIM (n - > ∞) [(f (x0+ α n) - f(x0- β n)] / ( α n+ β n) = f'(x0) | Math enthusiast | Mathematical art

Calculus derivative proof Let f (x) be differentiable at point x0, α n, β N is a positive sequence tending to zero, and it is proved that LIM (n - > ∞) [(f (x0+ α n) - f(x0- β n)] / ( α n+ β n) = f'(x0)

Just use the definition of derivative

Is the calculus of freshman similar to the derivative of senior three?

The derivative of senior three is very elementary. I didn't even elaborate on the limit. I just talked about the limit by saying "tend to a certain number". In fact, the limit is the cornerstone of calculus. Calculus is the process of calculating the limit. In college mathematics, the limit has a very strict description and proof from definition, operation to various formulas. The essence of derivative is one

Difference and relation between limit and derivative

Derivative studies the rate of change of function, and limit is the method of studying derivative

Relationship between limit and derivative Is there a limit and a derivative? Does a derivative have a limit? What's going on?

Is there a limit and a derivative? Not necessarily. You draw a broken line. There is a limit at the inflection point, but the left and right derivatives at the inflection point are different, so there is no derivative. Is there a limit when there is a derivative? Yes, the definition formula of derivative actually includes the term limit. In fact, from the image, the limit is only phase

If SiNx + cosx = 1 / 5, and 0

sinx+cosx=1/5.(1)
(SiNx + cosx) square = 1 / 25
1+2sinx*cosx=1/25
2sinx*cosx=-24/25
sinx*cosx-12/25.(2)
∵0

SiNx + cosx = root 2, TaNx + Cotx =? The straight line x + 7Y = 10 divides the circle x square + y square = 4 into two arcs. The absolute direct difference between the two arc lengths is equal to?

tanx+cotx＝（sinxsinx+cosxcosx）/sinxcosx
=1/sinxcosx
Because SiNx + cosx = root 2
So (SiNx + cosx) (SiNx + cosx) = 2
sinxcosx=0.5,
That is, TaNx + Cotx = 2

Calculus derivative proof Let f (x) be differentiable at point x0, α n, β N is a positive sequence tending to zero, and it is proved that LIM (n - > ∞) [(f (x0+ α n) - f(x0- β n)] / ( α n+ β n) = f'(x0)