1 The water-soluble vitamins (B vitamins and vitamin C) are absorbed easily along with the water in the small intestine, where they then travel through the body via the blood vessels. 2  Fat-soluble vitamins (vitamins A, D, and K) are absorbed just like the fat mentioned above.
A function f is said to be continuously differentiable if the derivative f'(x) exists and is itself a continuous function. Though the derivative of a differentiable function never has a jump discontinuity, it is possible for the derivative to have an essential discontinuity. For example, the function.
Suppose is a function of which is a function of (it is assumed that is differentiable at and, and is differentiable at. To prove the chain rule we use the definition of the derivative. We now multiply by and perform some algebraic manipulation. Note that as approaches 0, also approaches 0.