Question

I study mathematics-economics in the second year of a bachelor programme: I’m one month into the course “Analysis I”. I would like to get more familiar with certain subjects. For a reasonable answer for this Q&A, I’d like a formal definition, a simple example/proof and summary of the mentioned subject.

The subject is differentiability of one real function of one variable.

Answer 1

Definition:-

In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain. As a result, the graph of a differentiable function must have a (non-vertical) tangent line at each point in its domain, be relatively smooth, and cannot contain any breaks, bends, or cusps.

More generally, if x₀ is a point in the domain of a function f, then f is said to be differentiable at x₀ if the derivative f ′(x₀) exists. This means that the graph of f has a non-vertical tangent line at the point (x₀, f(x₀)). The function f may also be called locally linear at x₀, as it can be well approximated by a linear function near this point.

For a function of one variable, the derivative gives us the slope of the tangent line, and a function of one variable is differentiable if the derivative exists. For a function of two variables, the function is differentiable at a point if it has a tangent plane at that point. But existence of the first partial derivatives is not quite enough, unlike the one-variable case.

Let f R R be a function. The derivative of function f at a is f (x)-f(a) f(ah) f(a) lim h-0 f'(a) lim h х — а х-а provided the limit exists. If so, we say f is differentiable at a.

Differentiability of real functions of one variable A function fUcR R, defined on an open set U, is said to be differentiable ata E U if any of the following equivalent conditions is satisfied: The derivative f (a) = lin f(a+h) - f(a) . exists h h- 0 f(a+h) f(a) Lh There exists a real number L such that lim 0. The number L, when it exists, is equal to f (a) h-+0 g(h) There exists a function g : U CR> R such that f(a + h) = f(a) + f'(a)h + g(h) and lim = 0 h-0 h

Theorem:-

If f(x,y) has continuous partial derivatives fx(x,y) and fy(x,y) on a disk D whose interior contains (a,b), then f(x,y) is differentiable at (a,b).

Theorem:-

If f is differentiable at (a,b), then f is continuous at (a,b).

Example The squaring function given by f(x)=x is differentiable at x= 3, and its derivative there is 6. This result is established by calculating the limit as h approaches zero of the difference quotient of f(3): f(3h) - f(3) (3h)2 32 f'(3) = lim h-0 = lim 0 h h 96hh2 -9 = lim h-0 6h h2 lim (6 h) = lim h h-0 h-0 The last expression shows that the difference quotient equals 6 h when h 0 and is undefined when h 0, because of the definition of the difference quotient. However, the definition of the limit says the difference quotient does not need to be defined when h = 0. The limit is the result of letting h go to zero, meaning it is the value that 6 h tends to as h becomes very small: -1 The squaring function lim (6+ h) = 6 +0 = 6. h-+0 Hence the slope of the graph of the squaring function at the point (3, 9) is 6, and so its derivative at x = 3 isf'(3) = 6. More generally, a similar computation shows that the derivative of the squaring function at x a is f'(a) = 2a (а + h)2 — а? = lim h-0 f(ah) f(a) f'(a) = lim h-0 h a2 2ahh2 = lim h-+0 a2 2ahh2 = lim h-0 h h = lim (2a h) = 2a -+ 0