You cannot use the curve fitting toolbox to fit a model that has constraints on it like the ones you have. At least not directly.
Lets look at your model function however. The general model is:
y = a*exp(b*x) + c*exp(d*x)
Your constraints are that
y(0) = 0
y'(0) = 0
What do they mean in terms of the parameters?
y(0)=0 ==> a + b = 0
Or, we can write it as a = -b, essentially allowing us to eliminate b from the model completely. So now we can re-write the model in the form
y = a*exp(b*x) - a*exp(d*x)
Next, consider the slope boundary condition at x=0. Again, since exp(0) is 1, this reduces to:
y'(0) = 0 ==> a*b - a*c = 0
That is a problem however. Why? If we assume a is non-zero (as the problem becomes trivial if a is zero) then we now know that b == c. So we can reduce the model once more, into the form:
y(x) = a*exp(b*x) - a*exp(b*x) = 0
Does that seem problematic? It has a difference of two identical terms. So the ONLY exponential model that is a sum of exactly two exponentials of the form that you want, that also has y(0)=y-(0)=0, is the oddly trivial one:
y(x) = 0
So, can you solve this using the curve fitting toolbox? I suppose you might be able to do so, although I'm not sure how to enter a model that is constant at zero for all x.
I would strongly recommend you choose a model that does allow the required properties to hold, but also allows a fit to exist. Without seeing your data, it is very difficult to know what that model might be. For example, you might be able to use a very simple model of the form:
y(x) = a2*x.^2 + a3*x.^3 + a4*x.^4
etc. Don't go too high in the order of that model, as it will rapidly have numerical difficulties. You may need to consider scaling your data. A virtue of the model form I have suggested is it automatically forces y(0)=y'(0)=0.
My guess is that the data you have has relatively large x. So, how would I scale this problem? I would scale it as:
Now, solve the problem using the curve fitting toolbox by scaling x and y as:
xs = x/s;
ys = y/s^2;
ft = fittype('a2*x.^2 + a3*x.^3 + a4*x.^4','indep','x');
mdl = fit(xs(:),ys(:),ft);
Don't forget to unscale the coefficients afterwards. If you need a better recommendation for a model, you need to provide the data.