If changes(h) in x are continually reduced in relative size in a floating point operation, a size is reached when h+x =x exactly for x<>0, so (f(x+h)-f(x))/h=0/0. So the usual humanly imagined limit process is impossible in a real digital calculator. But 'the scale up' works for non zero x which require relatively few digits compared to calculator precision. Below, the number of digits typed for w must be small compared to the scale t so the didits for various terms of the polynomial expansion are separated by zeros, and the whole string of digits is within the browser's precision.

To stay close to binary logic I use rational numbers: x=w/s and w,s,t,n are integers.
Derivative of x^n ={ (w/s+ 1/(s*t) )^n - (w/s)^n}/(1/(s*t)).
    Factoring 1/(s*t) to outside, { (t*w + 1) +1)^n - (w*t)^n }*  [(1/(s*t))^n] / [1/(s*t)] .
Traditionally we 'find the limit as t is unbounded'. But in an integer calculator we only need t large enough to separate, with zeros, the strings of digits of the products.
To work with only positive integers, I reorder the divisions until they can be applied to sufficiently large numbers. Also t ,s for decimal work are to be a power of ten. This only limits x to finite approximations of infinite decimals. For example: 1/3 is approximated as close as desired well within calculator accuracy. 'well within' is to allow 0's between significant digits in the differentiation process. That is 1/3 is replaced by 333000/1000000=.333000 or if the calculator allows the desired powers of 333300001, use 33330000/100000000.
So we have { (t*w + 1)^n - (t*w)^n }/(s*t)^(n -1) = (Z^n-z^n)/(s*t)^(n-1) for z=t*w, Z=z+1.
This is very general since most physical functions can be generated or approximated by polynomials whose derivative is a sum of derivatives of each term.

Discussion

• This is just a simple explicit example of the general digital principle Digital transform=nor-formula. (10k): any transform of digital input to digital output is equivalent to a finite propositional logic formula common to computers. And arguments that cannot be put in a nor-formula are misleading. The only transform digital I/O that cannot be equaled by a nor-formula gives different(inconsistent) values at different times and time is not part of the input.
• A simple generalization from polynomials is for those functions that have 'separable' variables: f(t*w)=g(t)*h(w). The above in this notation: (t*w)^n = (t^n)*(w^n). So the more general calculator derivative is { f(w+1/t) - f(w) } / (1/t) = g(1/t) * t * { h(t*w+1) - h(t*w) } for t large enough to keep the digits due to the 1 out of the final result.
• Most computers represent decimals as 2 binary integers(floating point): digits and decimal place. Thus we can observe the relative change in a function f(x) relative to its argument x as this representation of the argument change is reduced.One could also increase the delta x from zero until the digits in x and delta x overlap - called significance - then continue this increase and note the plateau one may see before delta x = x; this plateau is the derivative. But if x=0, one must consider the computational nature of the function - what range of x values around 0 have significance - then find the plateau.
• A calculator proof of the 'fundamental theorem of calculus' should be similar.
• Legendre polynomials(LP) can be analysed by similar binary techniques. They are finite polynomials whose integration is simply anti-differentiation of x^n. That is, integral of x^n is (1/(n+1))*(x^(n+1)) using binary differentiation above. Thus one can easily demonstrate, binarily, that LP form an orthoginal set by anti-differentiation of the polynomials that result from the product of any 2 LP.
• The 'least square error fit' of a polynomial to any given finite set of points requires only binary differentiation with respect to the coefficients(an) of ( sum over i of (fi - sum over n of [an*x^n])^2 ). Then solve the simultaneous linear equations for these coefficients. All the multiplications and devisions of that solution is defined in nor-formulas in common computers.
• Number theory problems are close to explicit propositional logic since they involve integers which are equal to binary numbers and often use only simple formulas of addition and multiplication which are translated into nor-formulas for bounded values.