Equation of a tangent line to the graph of function `f` at point `(x_0,y_0)` is given by `y=f(x_0)+f'(x_0)(x-x_0).`
Since every tangent passes through the origin `(0,0)` we have
Let us write the equation using usual notation for differential equations.
Now we separate the variables.
Integrating the equation, we get
`ln y=ln x+ln c`
`c` is just some constant so `ln c` is also some constant. It is only more convenient to write it this way.
Taking antilogarithm gives us the final result.
There fore, our functions `f` have form `f(x)=cx` where `c in RR.`
Graphically speaking these are all the lines that pass through the origin. Since the tangent to a line at any point is the line itself the required property is fulfilled.
Graph of several such functions `f` can be seen in the picture below.