What is Differentiable: Definition and 284 Discussions

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. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp.
More generally, for x0 as an interior point in the domain of a function f, then f is said to be differentiable at x0 if and only if the derivative f ′(x0) exists. In other words, the graph of f has a non-vertical tangent line at the point (x0, f(x0)). The function f is also called locally linear at x0 as it is well approximated by a linear function near this point.

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  1. A

    Differentiable function, limits, sequence

    f is differentiable on (a,\infty) and \lim_{x\to\infty}\frac{f(x)}{x}=A I am trying to prove that there exists a sequence \{x_n\}, x_n\rightarrow \infty, such that f'(x_n)\rightarrow A. Any help would be appreciated.
  2. G

    Understand differentiable manifolds

    I am trying to understand differentiable manifolds and have some questions about this topic: We can think of a circle as a 1-dim manifold and make it into a differentiable manifold by defining a suitable atlas. For example two open sets and stereographic projection etc. would be the...
  3. R

    Integral of a continuously differentiable function on [a,b].

    suppose f is real, continuously differentiable function on [a,b], f(a)=f(b)=0 and integral f^2dx=1 show [integral(xf(x)f'(x)dx)= -1/2 over [a,b]
  4. C

    I have to prove x^3 is differentiable at x=4

    I have to prove x^3 is differentiable at x=4 using the definition of what it means for something to be differentiable. So I was wondering if I just have to show that f'(4) = \lim_{x \to 4} \frac {f(x) - f(4)}{x - 4} exists, where f(x) = x^3. So... f'(4) = \lim_{x \to 4} \frac {x^3 -...
  5. V

    Is |x|^3 differentiable?

    [SOLVED] Is |x|^3 differentiable? Homework Statement Is |x|^3 differentiable? Homework Equations Def: \ Let \ f \ be \ defined \ (and \ real-valued) \ on [a,b]. \ \ For \ any \ x \in [a,b], \ form \ the \ quotient \phi(t)=\frac{f(t)-f(x)}{t-x} \ \ \ \ (a<t<b, \ t\neqx), \\...
  6. A

    How is 1/f(x) differentiable at 0.

    I can prove derivative of 1/f(x) = -f'(x)/(f^2). But how can this be differentiable at x=0 (if f(x) is not equal to 0) as my class notes claim Thanks
  7. L

    Differentiable functions

    Homework Statement If f is differentiable at x=0 and g(x) = [f(x)]^2, f(0) = f'(0) = -1, then g'(0) = Homework Equations MC Answers: (A) -2 (B) -1 (C) 1 (D) 4 (E) 2 The Attempt at a Solution The only thing I could think of was that if g(x) = (f(x))^2 then g'(0) = (f'(0))^2...
  8. quasar987

    Real analysis - show convex functions are left &amp; right differentiable

    [SOLVED] Real analysis - show convex functions are left &amp; right differentiable Homework Statement Let f:R-->R be convex. Show f admits in every point a left derivative and a right derivative. Homework Equations A function f:R-->R is convex if x1 < x < x2 implies f(x)\leq...
  9. S

    Prove f(x) is differentiable at x=1 where f(x)=2x^2 x=<1, =4x-1 x>1

    prove f(x) is differentiable at x=1: f(x)=2x^2 x(less than or equal to)1 4x-1 x>1
  10. B

    Making a piecewise defined function differentiable

    I have to find the values of a and b in terms of c so that this function is differentiable. Attached is the problem and my work, but I think that there's an error somewhere in my attempt. Any advice?
  11. S

    Determing where function is differentiable (Complex Analysis)

    Homework Statement Determine where the function f has a derivative, as a function of a complex variable: f(x +iy) = 1/(x+i3y) The Attempt at a Solution I know the cauchy-riemann is not satisfied, so does that simply mean the function is not differentiable anywhere?
  12. J

    What does it mean for a function to be N-times differentiable?

    Sometimes I've encountered functions f:\mathbb{R}^n\to\mathbb{R}^m being called N-times differentiable. What does it mean, precisely? I know that for a function to be differentiable, if is not enough that the partial derivatives \partial_i f_j exist, but instead the derivative matrix Df must...
  13. M

    How do you normally check if functions are differentiable?

    please help me to find out the solution of this question its very simple but i am confused.the question is "CAN WE CHECK A FUNCTION WHICH IS NOT DIFFERENTABLE EXCATLY AT TWO POINT>IF YES THEN HOW WE CHECK IT" can i use sin or cos function,or is the graph is straight or not
  14. A

    Nowhere differentiable, continuous

    Hello need help with this one. f:[0,1] --> [0,1] f( .x1 x2 x3 x4 x5 ...) = .x1 x3 x5 x7 ( decimal expansion) prove that f is nowhere diffrentiable but continuous. i tried by just picking a point a in [0,1] and the basic definiton of differentiability about that point...doesnt seem...
  15. R

    Solving for m and b in a differentiable function at x=2.

    f(x)= x^4 x less than or = 2 mx+b x is greater than 2 Find the values of m and b that make f differentiable everywhere. so what i was trying to do was to find where the graph of x^4=mx where x=2 so that the mx+b function would start at where ever x^4 left off at x=2 i...
  16. D

    Definition of a differentiable function

    I need to know the definition of a differentiable function at a point in Banach spaces, my notes has a certain ambiguity and I can't find a book with the definition. Thanks.
  17. M

    Twice differentiable but not C^2

    I need to find an example of such a function. I know that x^2sin(1/x) is differentiable but not C^1, but I'm having trouble extending this to C^2.
  18. P

    Continuously Differentiable Piecewise Function?

    Here is a piecewise polynomial function: f(x) = x^2 + 1 if x <= 1 f(x) = 2x if x > 1 I need to prove that this function is differentiable at x = 1? It's a parabola that turns into a line. It doesn't have any gaps or corners. The limit of f(x) as x approaches 1 is 2, and the limit of...
  19. L

    Is f Differentiable? - Vector Calculus

    I have a problem regarding the function f (x,y) = {x*y*(x^2-y^2)/(x^2+y^2) if (x,y)!=(0,0) and f(x,y)=0 if (x,y)=(0,0). I am asked if this function is differentiable. Running it through a graphing program it looks differentiable. I know the partial derivatives of it in terms of x and y are...
  20. N

    What is the relationship between slope and symmetry in differentiable functions?

    :smile: For all real numbers x, f is a differentiable function such that f(-x) = f(x). Let f(p) = 1 and f'(p) = 5 for some p>0. a) Find f'(-p). b)FInd f'(0). c)If ß1 and ß2 are lines tangent to the graph of f at (-p,1) and (p,1) respectibely, and if ß1 and ß2 intersect at point Q, find the...
  21. L

    Solving a Real-Valued Differentiable Function Problem

    Alright, here is the problem. For all real numbers x, f is differentiable function such that f(x)=f(-x). Let f(p)=1 and f'(-p)=5, for some p>0 a) Find f'(-p) b) f'(0) c) If L1 and L2 are lines tangent to the graph of f at (-p,1) and (p,1) respectively, and if L1 and L2 intersect at...
  22. S

    Determine when f is differentiable

    How do you determine when f is differentiably from a real analysis standpoint (no graphs and calculus)? Would I simply look for a point of discontinuity? We have 4 problems on our homework assignment involving this issue and I don't see one example in my notes or the book adressing it. Here is...
  23. T

    What's the definition of a point being differentiable?

    "Suppose f is continuous on [a,b] and c in (a,b). Suppose f is differentiable at all points of (a,b) except possibly at c. Assume further that lim(x->c)f'(x) exists and is equal to k. Prove that f is differentiable at c and f'(c)=k" Since the lim f'(x) as x->c exists, f'(c) either equals k...
  24. R

    What is the Importance of Infinitely Differentiable Functions in Mathematics?

    infinitely differentiable doesn't care if all the higher derivatives are zeroes (like for polynomials), it only has to be defined...correct?
  25. R

    Closed Form Solutions for Differentiable Inverse Functions

    closed form?? let f:u \rightarrow R^n be a differentiable function with a differentiable inverse f^{-1}: f(u) \rightarrow R^n . if every closed form on u is exact, show that the same is true for f(u). Hint: if dw=0 and f^{\star}w = d\eta, consider (f^{-1})^{\star}\eta. i don't...
  26. L

    Proving Thomae's Function f(x) Not Differentiable"

    Thomae function f(x):(0,1)->R f(x)=p/q x is rational number (p and q are relatively prime natural number) f(x)=0 x in irrational number show that f is not differrentiable. l can show that this function is not differentiable at rational number. But i can't sequence that is example. not...
  27. U

    Can the line x=0 be differentiable?

    can the line x=0 be differentiable? the slope would be infinity right? so does that mean it is differentiable?
  28. G

    For what values of x is │x^2-4│ not differentiable?

    For what values of x is │x^2-4│ not differentiable? Is there a way to solve it without looking at the graph?
  29. S

    Continuous and nowhere differentiable

    The Harvey Mudd College Math dept presents the Weierstrass' function: f(x)=\sum_{n=0}^{\infty} B^nCos(A^n \pi x) as an example of a continuous function nowhere differentiable if 0<B<1 and AB>1+\frac{3\pi}{2}. Surely it converges to a continuous function if 0<B<1 regardless of the value of...
  30. J

    Twice continuously differentiable function

    Hello again, another problem: given: a function f:[0,\infty)\rightarrow\mathbb{R},f\in C^2(\mathbb{R}^+,\mathbb{R})\\ The Derivatives f,f''\\ are bounded. It is to proof that \rvert f'(x)\rvert\le\frac{2}{h}\rvert\rvert f\rvert\rvert_{\infty}+\frac{2}{h}\rvert\lvert...
  31. Q

    ? - infinitely differentiable solutions to initial value problems

    Hi, I am interested to know whether a theory exists that allows to answer the following sort of question. Does a solution of initial value problem of second order differential equation is infinitely differentiable on the set of positive real numbers? For example, 1) the solution of...
  32. L

    Prove that f(x) is not differentiable at x=0

    Hi I am a calc student in great need. If any1 can please help me thank u very much. Here it is For the func. f(x) = { 0 x < or/and = 0 2x +1 x > 0 Proove that f(x) is not differentiable at x=0 Also 2. A two piece ladder leaning against a wall is...
  33. D

    Why is a function not differentiable at a point?

    I'm only in high school, and I was wondering: Why are some functions not differentiable at certain points?
  34. C

    How to tell if a function is differentiable or not

    If f is a function defined by the fomula f(x)=xe^(modx), then show that f is differentiable at every point c, with f'(c)=(mod(c) +1)e^(modx) The hint that is given is 'consider separately the cases cgreater than 0, c less than 0 and c=0 To prove that f is differemtiable at...
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