- #1
sara_87
- 763
- 0
prove f(x) is differentiable at x=1:
f(x)=2x^2 x(less than or equal to)1
4x-1 x>1
f(x)=2x^2 x(less than or equal to)1
4x-1 x>1
cristo said:The derivative of a function at a point can be expressed as the limit of an expression. You should be able to get two limits; one for each branch of the function. If these are the same, then the function is differentiable at x=1.
PhY said:You Mean Integration.
Argh, My Integration is Rusty.
its the opposite of Differentiation.
so 2x would be x^2
2x^2x ...i don't know, because its F(X)^G(X).
You need to hear from somebody else on this.
Or more precisely, [tex]\frac{d}{dx}\int^x_a f(t) dt = f(x)[/tex] where a is a constant. "The Opposite of differentiation" is what people told me before I started integral calculus as well, and it screwed up my understanding a heap load.PhY said:Argh, My Integration is Rusty.
its the opposite of Differentiation.
2x^2x ...i don't know, because its F(X)^G(X).
You need to hear from somebody else on this.
Differentiability is a mathematical concept that describes the smoothness of a function. A function is differentiable at a point if it has a well-defined derivative at that point, which represents the rate of change of the function at that point.
To prove that a function is differentiable at a specific point, we need to show that the limit of the difference quotient (the difference between the function value at a point and a nearby point, divided by the distance between the two points) exists as the distance between the two points approaches zero. In other words, we need to show that the function is continuous and has a well-defined derivative at that point.
No, a function cannot be differentiable at a point if it is not continuous at that point. A function must be continuous at a point in order for the limit of the difference quotient to exist and for the function to have a well-defined derivative at that point.
A function is differentiable if it has a well-defined derivative at a point, while a function is continuously differentiable if it has a well-defined derivative at every point in its domain. In other words, a continuously differentiable function is also differentiable, but the converse is not necessarily true.
To prove that a specific function is differentiable at a specific point, we can use the definition of differentiability and apply the limit definition of the derivative. We need to show that the difference quotient approaches a finite value as the distance between the two points approaches zero. We can also use the rules of differentiation to show that the function has a well-defined derivative at that point.