Hello there. To solve this question, we'll have to remember some properties about functions and derivatives.
We want to determine any possible functions h and g, where h cannot equal g, but knowing that h is the derivative of g and g is the derivative of h.
Mathematically, we have that

The way we can solve this is to consider that both h and g are twice differentiable, hence you differentiate both sides of the first expression to get:

But knowing g'(x) is h(x), we get

Subtract h(x) on both sides of the equation

This is a second order linear homogeneous differential equation.
To solve it, assume that

For a fixed constant λ
Taking its derivative twice, we get

Plugging this into our equation, we get

Factor the exponential term

Since we know that

We get that

It is called the characteristic equation for this differential equation.
Solving it gives us:

Therefore the solutions to this equation are

But since this is a linear equation, any finite combination of these solutions are solutions to it, therefore we say that

Is the general solution to this equation for any real constants c1 and c2.
Taking the derivative of this function, we'll see that

And the only case in which g(x) = h(x), that is what we don't want to have, is when

So it is possible to find infinitely many functions h and g of x satisfying this condition, as long c_2 is not equal to zero.
Take, for example, the solution:

They are not equal, but satisfies the properties:

As wanted.
Another way of solving is: Add the conditions as follows

Now make the following substitution:

Since the differential operator is linear, it is true that

Hence we have that

This is a first order linear homogeneous differential equation, but it is also a separable equation, that we can solve as:
Divide both sides of the equation by f(x) (knowing it is not identically equal to zero)

Integrate both sides with respect to x, such that you get

The left hand side of the integral equation can be calculated knowing that

Hence we get

For the right hand side, we have the integral of a power of x. Remember that

Therefore applying the power rule:

We get that

Hence the equation gives us

Subtract C on both sides of the equation

K_1 is another arbitrary constant.
Raise both sides of the equation as a power of e, as follows:

Here we applied the property of powers:

Since e^(K_1) is another constant, we called it K.
In this case, since it is a modular equation for f(x), we say that K must be a positive constant.
Returning to the definition, we see that

Making g(x) = h'(x), we have another equation

This time it is not a homogeneous equation, although it can be solved in a very similar way. The solution will be

Just as we found before, for constants A and B.
We perform the same argument to say that h(x) will only be equal to g(x) if B = 0, therefore it is possible to find functions of this form satisfying these properties.