Pythagorean theorem: the square of the hypotenuse of a right triangle is equal to the sum of the squares of two right angles.
2. Mean value theorem: If the function f(x) is continuous in the closed interval [a, b] and derivable in the open interval [a, b], there exists ξ∈(a, b), so that f' (ξ) = f (b)-f (a)/ (.
3. Lagrange mean value theorem: If the function f(x) is continuous in the closed interval [a, b] and derivable in the open interval [a, b], there exists ξ∈(a, b), so that f (b)-f (a) = f' (ξ) (b).
4. Rolle's Theorem: If the function f(x) is continuous in the closed interval [a, b] and derivable in the open interval [a, b], and f(a)=f(b), there exists ξ∈(a, b), so that f'(ξ)=0.
5. Cauchy mean value theorem: If the functions f(x) and g(x) are continuous in the closed interval [a, b] and derivable in the open interval [a, b], there exists ξ∈(a, b), which makes f (b) g' (ξ)-f'.
6. Newton-Leibniz formula: If the function f(x) is continuous in the closed interval [a, b] and derivable in the open interval [a, b], then ∫ _ a BF (x) dx = f (b)-f (a) = f (b)-f (.
7. Taylor's Theorem: If the function f(x) is the derivative of order n at point A, then for any real number x, there is f (x) = f (a)+f' (a)+f' (ξ) (x-a) 2/2! +...+f^n'(ξ)(x-a)^n/n！ , where ξ∈(a, x).
8. Robida's Law: If the functions f(x) and g(x) both approach infinity or infinitesimal at point X, when the independent variables approach this point, their limit forms are all 0/0, ∞ /∞, 0*∞ or ∞ -∞, then the limit problem can be solved by derivation.