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. M

    When is a function not differentiable?

    I'm curious about the conditions for when a function f(x) is not differentiable
  2. B

    Piecewise Differentiable Equation?

    Homework Statement K and M are constants. If h is differentiable at x=2, what are the values of k and m. h(x)= kx^2 + 1, 0<x<2 mx - 3, 2<x<5 All of the "<" signs are "less than or equal to" Homework Equations Not sure The Attempt at a Solution I tried setting the...
  3. S

    Is this function differentiable?

    Hey guys, I'm just wondering if I got this question right: Discuss where the following in R^{2} is differentiable: f(x,y)=(x^{3}+y^{3})^{2/3} So I take the partial derivative: f_{x}(x,y)=\frac{2x^{2}}{(x^{3}+y^{3})^{1/3}} and see that f(x,y) might not be differentiable at (0,0), so...
  4. T

    Partial derivative; is the function differentiable

    Homework Statement http://dl.dropbox.com/u/907375/Untitled.jpg Homework Equations Δz = f(a + Δx, b + Δy) - f(a, b) [PLAIN][PLAIN]http://dl.dropbox.com/u/907375/Untitled2.jpg The Attempt at a Solution f_x(0,0)=lim(h->0)=0 f_y(0,0)=lim(h->0)=0 f(x,mx)=lim(h->0)=0...
  5. S

    Is mod x differentiable at all points

    f(x)= mod x is this function differentiable at all points other than 0.
  6. A

    How to prove differentiable everywhere?

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  7. R

    If Partial derivatives exist and are continuos then function is differentiable

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  8. B

    How cubed root of x is not differentiable at 0

    I know that it isn't. I just want to know how I could, step by step, prove that this function is not differentiable at x=0.
  9. R

    Gradient (dot) cross product of 2 differentiable vector functions

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  10. S

    Proving the Differential of $\det (A)$ with Differentiable Elements of t

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  11. L

    What is a Differential Structure on a Manifold?

    Hi, I just started learning differential geometry. Got some questions. Thanks in advance to anyone who can help! Consider the one-dimensional manifold represented by the line y = x for x<0 and y = 2x for x>= 0. Now if I consider the altas with two charts p(x, y)=x for x<-1 and q(x,y)=y for...
  12. N

    Finding a formula from a single variable differentiable function

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  13. T

    Proving Non-Differentiability of a Function at a Specific Point

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  14. D

    A function bounded and differentiable, but have an unbounded derivative?

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  15. D

    Is the Function f(x) Differentiable at x=0 and x=1?

    f(x)= x=1/x-1 if x ≤ 0 x^2-2x +1 if 0 < x < 1 ln x if x ≥ 1 Here's what I have so far lim f(x)= lim x+1/x-1= -1, LHL= -1 x →0- x→0- lim f(x)= lim x^2-2x+1= (1/2)^2 - [(2)1/2] +1= 1/4, RHL= 1/4 x→0+ x→0+...
  16. S

    Understanding Twice Differentiable Functions with f''(x) ≥ 0

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  17. H

    What is the domain? Where is it differentiable? What is the derivative?

    I'm private studying a section in Linear Algebra first dealing with complex numbers. Now, I am ok with most of the answers, but I need help particularly with where this function is differentiable. Homework Statement Let f(z) = ln|z| + i arg z, 0<=arg z<2pi. What is the domain? What is...
  18. M

    Which differentiable functions R to R are bijective?

    I innocently gave my students a problem: Which differentiable functions f: R \rightarrow R are bijective? "Innocently", I say, because I'm finding it hard to come up with any simple set of conditions that are both necessary and sufficient. Here's what I can say so far: (1) If f'(x) \neq 0...
  19. P

    Is x/(1+e^(1/x)) differentiable at x=0

    Homework Statement is x/(1+e1/x) differentiable at x =0 Homework Equations The Attempt at a Solution i calculated the right and left hand derivative and got them as 0 and 1 respectively , so it should not be differentiable, is it right my book says that it is differentiable ?
  20. S

    If f+g is differentiable, then are f and g differentiable too?

    Homework Statement if f and g are two continuous functions and f+g is differentiable...are f and g differentiable? if not give a counter example! Homework Equations The Attempt at a Solution
  21. M

    F differentiable proves continuity

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  22. S

    Is a this function differentiable?

    Hi all, I was just wondering if a function that is continuous and differentiable for all xεR, but where the domain is restricted to closed interval [a,b], does the derivative exist at x=a or x=b?
  23. P

    Inflection point of non continuous or non differentiable function

    Homework Statement three functions: y=\begin{cases}\arctan \frac{1}{x}\ x\neq0\\ 0\ x=0\end{cases} y=\frac{1}{x}, y=|x^2-1| and what about inflection point? The Attempt at a Solution first function is concave on left of 0, convex on right, so from definition it should be inflection point...
  24. C

    Finding Continuous & Differentiable Points of f in {R}^3

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  25. G

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  26. S

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  27. R

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  28. K

    Finding a Point of Equality in a Twice Differentiable Function on [0,1]

    Let g:[0,1]-->R be twice differentiable(both g and g' are differentiable functions) with g''(x)>0 for all x in [0,1]. If g(0)>0 and g(1)=1, show that g(d)=d for some point d in (0,1) iff g'(1)>1. I thought I might use the MVT. g'(c)=g(1)-g(0)/1=1-g(0) g'(c)<0 then
  29. C

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  30. T

    Tangent and Normal Spaces, lagrange multiplier and Differentiable Manifolds question.

    Homework Statement OK I have a Differential Calculus exam next week and I do not understand about Differential Manifolds. We have been given some questions to practise, but I have no idea how to do them, past a certain point. For example 1. Study if the following system defines a manifold...
  31. C

    Showing f is Differentiable at c: A Challenge

    Homework Statement Let I be an interval, and f: I --> R be a continuous function that is known to be differentiable on I except at c. Assume that f ' : I \ {c} --> R admits a continuous continuation to c (lim x -> c f ' exists). Show that f is in fact also differentiable at x and f ' (c) =...
  32. atomqwerty

    Tangent Space and Manifold of a Cubic Surface

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  33. M

    Prove that the functin is differentiable at (0, ,0).

    Homework Statement Let r>0, and let f be a function from B_{r}(\textbf{0}) \rightarrow \textbf{R} , and suppose that there exists an \alpha > 1 such that |f(\textbf{x})| \leq ||\textbf{x}||^{\alpha} for all \textbf{x} \in B_{r}(\textbf{0}). Prove that f is differentiable at 0. What happens...
  34. J

    Differentiable / continuous functions

    Homework Statement give an example of a function f: R --> R that is differentiable n times at 0, and discontinous everywhere else. Homework Equations ---The Attempt at a Solution i got one, and i proved everything, i just want to make sure what i did is correct: f:x n+1 when x is rational...
  35. Z

    Find the points where the function is not differentiable

    Homework Statement Find the points where the function given by is not differentiable. The Attempt at a Solution I got the doubtful points as +-1, 2 How do I check the differentiability now? The mod. function is confusing me a bit.
  36. Z

    Find the points at which the function is not differentiable

    Homework Statement Find the points at which the function is not differentiable. Homework Equations It is not asked to check differentiability at a particular point. How do I find the points which are not differentiable? The function is not continuous at x=0
  37. J

    Function differentiable, but derivative not bounded

    Homework Statement Give an example of a function f that is differentiable on [0,1] but its derivative is not bounded on [0,1] Homework Equations The Attempt at a SolutionOk, I know that the derivative f' cannot be continuous, because then it would be bounded on [0,1]. I also know that it...
  38. K

    Linear independence with differentiable functions

    I don't this this is an overly complicated proof but it is one I have never seen or done before. Let f be a polynomial with atleast two non-zero terms having different degrees. Prove that the set {f(x),xf'(x)} is linearly independent in P Proof: With out loss of generality we can...
  39. J

    So the correct answer isdy/dx = (-sin4x - 4siny) / (4y + 4cos4x)

    Use implicit differentiation to find dy/dx. y is a differentiable function of x 2y^2+4xsiny = cos4x Here is what I did: 4y*dy/dx + 4siny+ 1/cosy*dy/dx = -sin4x + 4 4y*dy/dx + 1/cosy*dy/dx = -sin4x - 4siny + 4 dy/dx(4y + 1/cosy) = -sin4x - 4siny + 4 dy/dx = (-sin4x - 4siny + 4) /...
  40. H

    Lie Algebra differentiable manifold

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  41. J

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  42. Telemachus

    Calc III - is this differentiable in all points?

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  43. T

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  44. E

    Projection of a differentiable manifold onto a plane

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  45. D

    Differentiable implies continuous

    ehhh
  46. K

    Show f(x) = { x/2 if x rational , x if x irrational is not differentiable at 0

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  47. K

    If f is differentiable at x = a, evaluate lim[h->0] (f(a+2h)-f(a+3h))/h

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  48. K

    Passing the limit through the derivative of a differentiable function

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  49. M

    Example of a differentiable structure on R

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  50. C

    Is f(x) differentiable at x=1?

    Given http://www.mathhelpforum.com/math-help/vlatex/pics/105_fde5ac6b051b4fac473487c7b4afa9e5.png Is f(x) differentiable at x=1? I know that we have to prove http://www.mathhelpforum.com/math-help/vlatex/pics/65_6fae3c52eaa96aaafdf2c225a900ea48.png exist/does not exist at x=1. But...
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