Does such a function as f(x)=f'(x) exist?

[Andrax]

i was wondering if this function actually exist or dosent if it does can someone provide an example

 

[slider142]

Have you tried f(x) = 0 ? :)
More seriously, there is a well-known function with this property: $f(x) = e^x$ .

 

[Vorde]

And to show yourself it, try solving the differential equation $\frac{dy}{dx} = y$

It doesn't really prove much, but its a nice way to convince yourself that what slider142 said is true!

 

[micromass]

Or more general: $f(x)=Ce^x$ for an arbitrary constant $C$. These are all the functions that satisfy your equation.

 

[Andrax]

Thanks we haven't gotten to logarithms yet

 

[HallsofIvy]

Then why did you ask this question?

 

[rbj]

it seems odd, pedagogically, that one would be learning about differential equations and calculus without first learning about logarithms.

 

[Bipolarity]

My guess is that the question arose out of OP's curiosity rather than actually being introduced to differential equations in the coursework. I remember when learning calculus I had similar curiosities myself, before even learning about exponential and logarithmic functions.

BiP

 

[Redbelly98]

It sounds like the OP is in the first semester of calculus, where at some point you are eventually told that e^x has the very property being discussed here. Probably the class hasn't gotten there just yet.

Also, the OP may have meant that they haven't covered logarithms yet in the calculus course. At least that would make more sense then never having gotten to them, ever.

 

[h6ss]

It might be interesting to note that some functions are also equal to their own nth derivatives, for example:

$\frac{\mathrm{d}^2}{\mathrm{d}x^2} e^{-x} = e^{-x}$

$\frac{\mathrm{d}^3}{\mathrm{d}x^3} e^{\frac{-x}{2}}\sin(\frac{\sqrt{3}x}{2}) = e^{\frac{-x}{2}}\sin(\frac{\sqrt{3}x}{2})$

$\frac{\mathrm{d}^4}{\mathrm{d}x^4} \sin x = \sin x$