Are there any 'lost' theorems of calculus that could be used to 'simplify' it? For example, are there ways to calculate derivatives without using limits, maybe by some forgotten methods in calculus?
As a matter of fact, there was something now called "lost calculus" or "algebraic calculus" in the 17th century, that avoided concepts like limits or infinitesimals, which where problematic at the time. It was developed by Descartes, Hudde and others, and is described in Suzuki's award winning paper The Lost Calculus (1637-1670): Tangency and Optimization without Limits. However, it only applied to algebraic functions, and it "simplifies" calculus only in the sense of avoiding more abstract concepts, rather than necessarily making computations simpler. So it was abandoned in favor of more general calculus of Newton and Leibniz based on infinitesimals, and later formalized using limits.
Descartes first introduced the idea in La Geometrie in 1637, and later simplified it to "the method of tangents" in 1638. Suppose we want to find the slope of the tangent to $y=x^2$ at $x=1$. The general equation of a line passing through $(1,1)$ is $y-1=m(x-1)$, and $m$ is the slope we are looking for. Since nearby lines intersect the graph at two points and the tangent only touches it at one algebraically the system
$$y=x^2,\ \ y-1=m(x-1)$$ must have a double root at $x=1$. Eliminating $y$ and factoring we get $$x^2-1-m(x-1)=(x-1)(x+1-m)=0,$$ so for $x=1$ to be a double root we must have $1-m=-1$ or $m=2$, which is the sought slope of the tangent.
This approach works for any polynomial $y=p(x)$, and more generally for rational and even algebraic functions given implicitly. For example, to find slope of the tangent at $x=a$ we write $y-p(a)=m(x-a)$ and look for the value of $m$ that makes $a$ a double root of $p(x)-p(a)=m(x-a)$. This only requires long division of $p(x)-p(a)$ by $x-a$, no limits or infinitesimals, and if the quotient is $q(x)$ then $m=q(a)$. To make the method computationally viable an algorithm more efficient than long division is needed for detecting double roots. Such method was provided by Jan Hudde, a talented Dutch mathematician who had to abandon mathematics for politics to save Netherlands from Spanish invasion, in two letters included into 1659 edition of Descartes' La Geometrie. It involves a clever modular reduction of polynomials that anticipates methods of modern algebraic geometry.