Calculus
Chain Rule
Differentiate composite functions by peeling layers
Mean Value Theorem
There exists a point where the instantaneous rate of change equals the average rate of change
Fundamental Theorem of Calculus
Integration and differentiation are inverse operations
Taylor's Theorem
Approximate smooth functions by polynomials with explicit error bounds
L'Hôpital's Rule
Evaluate indeterminate-form limits by differentiating numerator and denominator
Intermediate Value Theorem
A continuous function on a closed interval hits every value between its endpoints
Rolle's Theorem
Between two equal values of a differentiable function there exists a point with zero derivative
Cauchy's Mean Value Theorem
A two-function generalization of the Mean Value Theorem relating derivative ratios to function value ratios
Darboux's Theorem
The derivative of a differentiable function satisfies the intermediate value property, even when not continuous
Implicit Function Theorem
A smooth equation F(x,y)=0 with invertible partial derivative in y locally defines y as a smooth function of x
Inverse Function Theorem
A smooth map with invertible derivative at a point is locally a diffeomorphism
Extreme Value Theorem
A continuous function on a compact set attains its maximum and minimum values
Weierstrass Approximation Theorem
Every continuous function on a closed bounded interval can be uniformly approximated by polynomials