Sage-Tips

What is uniformly differentiable?

Differentiability. Uniform differentiability of a function f also is a condition on its variation: it factors as f(y) f(x) & F(x, y)(y x), where F(x, y) # F(x, x) as y # x. If f is uniformly differentiable, its derivative is the function f'(x) & F(x, x).

What determines if a function is differentiable?

A function is said to be differentiable if the derivative of the function exists at all points in its domain. Particularly, if a function f(x) is differentiable at x = a, then f′(a) exists in the domain.

What is the difference between differentiable and derivable?

Differentiability refers to the existence of a derivative while differentiation is the process of taking the derivative. So we can say that differentiation of any function can only be done if it is differentiable.

What does it mean if something is differentiable calculus?

A function is differentiable at a point when there’s a defined derivative at that point. This means that the slope of the tangent line of the points from the left is approaching the same value as the slope of the tangent of the points from the right.

Is derivative function differentiable?

In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain.

How do you know if a complex function is differentiable?

‌ Let f:A⊂C→C. The function f is complex-differentiable at an interior point z of A if the derivative of f at z, defined as the limit of the difference quotient f′(z)=limh→0f(z+h)−f(z)h f ′ ( z ) = lim h → 0 f ( z + h ) − f ( z ) h exists in C.

Is the inverse of a differentiable function differentiable?

The inverse function theorem states that if you have a differentiable function who’s derivative is non-zero, then locally the inverse function exists and is differentiable. For notational ease, call g(x) the inverse function of f, then you get g′(x)=1f′(g(x)).

What are some examples of non-differentiable functions?

Generally the most common forms of non-differentiable behavior involve a function going to infinity at x, or having a jump or cusp at x. There are however stranger things. The function sin(1/x), for example is singular at x = 0 even though it always lies between -1 and 1.

How do you define differentiability?

So far, we have an informal definition of differentiability for functions f: R 2 → R: if the graph of f “looks like” a plane near a point, then f is differentiable at that point.

How do you prove a function is differentiable at x = a?

In single variable calculus, a function f: R → R is differentiable at x = a if the following limit exists: f ′ ( a) = lim x → a f ( x) − f ( a) x − a. This limit exists if and only if lim x → a ( f ( x) − f ( a) x − a − f ′ ( a)) = 0.

Is f(x) = x y + 2x + y differentiable at 0?

We had previously used our informal definition of differentiability to determine that the function f ( x, y) = x y + 2 x + y is differentiable at ( 0, 0). Let’s verify this using our new, formal definition of differentiability. We’ll show that the function f ( x, y) = x y + 2 x + y is differentiable at ( 0, 0).