When does the derivative equal the integral?

We find a function such that its antiderivative equals its derivative #shorts

#calculus,#derivative,integral,numbers,integration,differentiation,derivative,integral,math elite,mathelite,maths,mathematics

[MathElite]

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[FreeGroup]

sin(ix) '=icos(ix)
icos(ix)' = sin(ix)

[nikodempatrycjuszswiercz]

Very nice idea! I completely forgot you could do that :).
You can also get more solutions by:
0. f(x) = f ' '(x) /// we see everything up to this in the video
1. multiply both sides by f '(x)
2. integrate (on both sides you can substitute for the function with fewer derivative symbols), multiply by 2 and write constants on the side of (f(x))^2 as one
2.5: should get (f(x))^2 + C = (f '(x))^2
3. undo squaring (+ / - square root both sides), i'll continue only with the positive part for simplicity
4. divide by the left side of the equation
4.5: we get 1=(f '(x)) / sqrt((f(x))^2 + C)
5. divide the problem into cases: C > 0, C = 0, C < 0
6. Integrate both sides (we can do this by substitution, similarly to step 2, u=f(x)), write both new constants on one side and as one letter (remember: not the same as C)
7. solve the resulting equation for f(x)
Ad.: Some functions end up being the same, but not all of them.

[LearnwithTejeshwar]

I found your mistake
F'(x) = re^rx
F"(x)= r^2.e^rx
F"(x) - F'(x) =0
r^2.e^rx - re^rx =0
(r^2 - r).e^rx =0
r( r-1 ).e^rx =0
r = 0, 1
So y = ce^0.x + ce^1.x
y = c + ce^x <- this is the answer
If I am wrong please tell me
And I appreciate your effort 👍

[sidkt]

Won’t Asinh(x) Bcosh(x) also work as it’s derivative and integral is Acosh(x) + Bsinh(x)?

[Ahmed-vsui]

How about -cos(-x) isnt the derivative sin(-x) and the integral sin(-x) ???

[TA]

why f(x) = e to the power of rx?(in the fourth line of the answer)