Search results
Results from the WOW.Com Content Network
In particular, any differentiable function must be continuous at every point in its domain. The converse does not hold: a continuous function need not be differentiable. For example, a function with a bend, cusp, or vertical tangent may be continuous, but fails to be differentiable at the location of the anomaly.
Differentiable programming is making significant strides in various fields beyond its traditional applications. In healthcare and life sciences, for example, it is being used for deep learning in biophysics-based modelling of molecular mechanisms.
In mathematics, the Weierstrass function, named after its discoverer, Karl Weierstrass, is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is also an example of a fractal curve .
For example, suppose that : is a differentiable function of variables , …,. The total derivative of f {\displaystyle f} at a {\displaystyle a} may be written in terms of its Jacobian matrix, which in this instance is a row matrix:
In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let () = (), where both f and g are differentiable and ()
For example, consider the function g(x) = e x. It has an inverse f(y) = ln y. Because g′(x) = e x, the above formula says that = =. This formula is true whenever g is differentiable and its inverse f is also differentiable. This formula can fail when one of these conditions is not true.
For functions of a single variable, the theorem states that if is a continuously differentiable function with nonzero derivative at the point ; then is injective (or bijective onto the image) in a neighborhood of , the inverse is continuously differentiable near = (), and the derivative of the inverse function at is the reciprocal of the derivative of at : ′ = ′ = ′ (()).
Under mild conditions (for example, if the function is a monotone or a Lipschitz function), this is true. However, in 1872, Weierstrass found the first example of a function that is continuous everywhere but differentiable nowhere. This example is now known as the Weierstrass function. [15]