Differentiable Definition and 290 Threads

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. D

    If f is infinitely differentiable and analytic on a dense set is f analytic?

    Let f: R->R. If f is infinitely differentiable and analytic on a dense set is f analytic? Is this true if we restric f to [0,1]? note: by analytic I mean the radius of convergence of the taylor expansion is non-zero about every point. Maybe this is simple but I was thinking about it and...
  2. M

    Converg. Seq. of Functions, Derivatives Bounded, Limit not Differentiable

    Homework Statement Find a sequence of differentiable functions $f_n\colon [a,b]\rightarrow\mathbb(R)$ s.t.: --there exists $M>0$ with $|f_n'(x)|\leq M$ for all $n\in\mathbb{N}$ and $x\in[a,b]$; --for all $n\in\mathbb{N}$, $|f_n(a)|\leq M$; --$(g_n)$ is a convergent subsequence with...
  3. A

    Show that f Uniform Differentiable implies f' Uniform Continuous

    Homework Statement A function f:(a,b)\to R is said to be uniformly differentiable iff f is differentiable on (a,b) and for each \epsilon > 0, there is a \delta > 0 such that 0 < |x - y| < \delta and x,y \in (a,b) imply that \left|\frac{f(x) - f(y)}{x - y}-f'(x)\right| < \epsilon. Prove that...
  4. F

    Infinitely differentiable vs. continuously differentiable vs. analytic?

    Hello. I am confused about a point in complex analysis. In my book Complex Analysis by Gamelin, the definition for an analytic function is given as :a function f(z) is analytic on the open set U if f(z) is (complex) differentiable at each point of U and the complex derivative f'(z) is...
  5. C

    Finding the Value of a Derivative with Given Function and Derivative Values

    Homework Statement Suppose u and v are differentiable functions of x. Use the given values of the functions and their derivatives to find the value of the indicated derivative. u(1)=2, u'(1)=-7, v(1)=7,v'(1)=-2 d/dx (uv) at x =1 Homework Equations The Attempt at a Solution
  6. M

    Is f(x) Differentiable at x = 1?

    let f(x) = 2-x if x<= 1 x^2 - 2x + 2 if x > 1 Is f diff at x = 1? At first I would say yes because f(x) is continuous at x = 1. But when I graph f '(x) it is obvious that the function is not differentiable at x = 1. My questions is... is there another way to determine if...
  7. M

    When is a function not differentiable?

    I'm curious about the conditions for when a function f(x) is not differentiable
  8. 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...
  9. 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...
  10. 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...
  11. S

    Is mod x differentiable at all points

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

    How to prove differentiable everywhere?

    Homework Statement See photo, part b and c Homework Equations The Attempt at a Solution For part b It seems it is trival, in part a we have proved that f_{x} and f_{y} exist. Obviously, they are differentiable for x and y\neq0 For part c. It seems there are 2 method to do it...
  13. R

    If Partial derivatives exist and are continuos then function is differentiable

    Homework Statement Hi I'm just looking for a link to the proof of this theorem: if the partial derivatives of function f exist and are continuous at a point then the function is differentiable there Or even the name would be helpful Its not a homework assignment per say, just something...
  14. 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.
  15. R

    Gradient (dot) cross product of 2 differentiable vector functions

    [b]1. For two differentiable vector functions E and H, prove that (Delta (dot) (e X h) = H (dot) (delta X e) - e (dot) (Delta X h) [b]2. Cross product and dot product. The Attempt at a Solution First I took did the left side of the equation, I took the cross product of vectors e and...
  16. S

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

    Homework Statement PROVE: If A(t) is nxn with elements which are differentiable functions of t Then: \frac{d}{dt}(det(A))=\sumdet(Ai(t)) where Ai(t) is found by differentiating the ith row only. Homework Equations I know I should prove this by induction on n The Attempt at a...
  17. 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...
  18. N

    Finding a formula from a single variable differentiable function

    Homework Statement Suppose that f is a differentiable function of a single variable and F(x,y) is defined by F(x,y) = f(x^2 - y) Problem: Given that F(0,y) = sin y for all y, find a formula for F(x,y) Homework Equations The Attempt at a Solution This is what the tutor had put...
  19. T

    Proving Non-Differentiability of a Function at a Specific Point

    Hello! I got the following function: \int_{0}^{x} \left(-1 \right)^{\lfloor 2^{i} \cdot t \rfloor} \ dt, \quad x \in \left[0,1 \right] I want to show it is not differentiable at x= 2^{-i} k where k is a natural number greater equal 0. I already calculated the right derivate by...
  20. D

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

    Can a function f: (a,b) in R be bounded and diffferentiablle, but have an unbounded derivative. I believe it can, but can not think of any examples where this is true. Anyone have any ideas?
  21. 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+...
  22. S

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

    Homework Statement 1. if a function is twice continuously differentiable with f''(x) >= 0 for all real values of x then (f(-x) + f(x))/2 >= f(0) ? 2. if a function is twice continuously differentiable with f''(x) >= 0 for all real values of x then tf(x) + (1-t)f(y) >=...
  23. 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...
  24. 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...
  25. 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 ?
  26. 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
  27. M

    F differentiable proves continuity

    Homework Statement If f is differentiable at x then f is continues at x Any help would be great. Homework Equations MUST USE epsilon delta definition to prove The Attempt at a Solution
  28. 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?
  29. 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...
  30. C

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

    Homework Statement Find the continuous points P and the differentiable points Q of the function f in {R}^3, defined as f(0,0,0) = 0 and f(x,y,z) = \frac{xy(1-\cos{z})-z^3}{x^2+y^2+z^2}, (x,y,z) \ne (0,0,0). Homework Equations The Attempt at a Solution If you want to look at the limit I'm...
  31. G

    Complex Analysis-Difference between Differentiable and Analytic

    Homework Statement Show that f(z) = x^2 + i(y^2) is diff at all points on y=x. Then show that is not analytic anywhere.Homework Equations Cauchy Riemann equations: fy = ifx <=> function is differentiable (I'm still unclear about the implications of CR-equations. It says in my book that if f is...
  32. S

    Prove differentiable implies continuous at x=xo

    1. Prove f is differentiable at x=xo implies f is continuous at x=xo using epsilon and delta notation. 2. I have gotten this far: absolute value(f(x)-f(xo)) <= absolute value(x-xo)*(epsilon + absolute value(f '(xo))) <= means less than or equal to. 3. I need to get here: absolute...
  33. R

    F(y)=summation 1/(y^2+m^2) is not differentiable.

    Homework Statement Is f(y)=\sum_{m=1}^\infty \frac{1}{y^2+m^2} differentiable? Homework Equations The Attempt at a Solution From the graph, it is obvious that f is not differentiable at y=0, but I don't know how to prove that. I proved that \sum_{m=1}^n f_m=\sum_{m=1}^n\frac{1}{y^2+m^2}...
  34. 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
  35. C

    Approximation of continuous functions by differentiable ones

    Homework Statement Let f: R-->R be continuous. For δ>0, define g: R-->R by: g(x) = (1/2δ) ∫ (from x-δ to x+δ) f Show: a) g is continuously differentiable b) If f is uniformly continuous, then, for every ε>0, there exists a δ1>0 such that sup{∣f(x) - g(x)∣; x∈R} < ε for 0<δ≤δ1The Attempt at...
  36. T

    Does the System Define a Manifold and How to Find Tangent and Normal Spaces?

    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...
  37. 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) =...
  38. atomqwerty

    Tangent Space and Manifold of a Cubic Surface

    Homework Statement In which points the surface \{\left(x,y,z\right)\in\Re^{3}|x^{3}-y^{3}+xyz-xy=0\right\} is a differentiable manifold (subvariedad diferenciable in spanish). Calculate its tangent space in the point (1,1,1). Homework Equations NA The Attempt at a Solution I've...
  39. 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...
  40. 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...
  41. 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.
  42. 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
  43. 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...
  44. 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...
  45. 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) /...
  46. H

    Lie Algebra differentiable manifold

    Okey, I have problem with the foundation of lie algebra. This is my understanding: We have a lie group which is a differentiable manifold. This lie group can for example be SO(2), etc. Then we have the Lie algebra which is a vectorspace with the lie bracket defined on it: [. , .]. This...
  47. J

    How to prove if f(x) is infinitely differentiable

    Say I have the function f(x) = x^5ln(x) I can differentiate f(x) say 6 times and I'm left with just ln(x). But this is not a proof! How do I go about proving it? Could I use the Taylor expansion?
  48. Telemachus

    Calc III - is this differentiable in all points?

    Homework Statement Hi there. I got next problem, and I must say if it is differentiable in all of its domain. f(x,y)=\begin{Bmatrix} (x+y)^2\sin(\displaystyle\frac{\pi}{x+y}) & \mbox{ if }& y\neq{-x}\\0 & \mbox{if}& y=-x\end{matrix} So, I thought trying with the partial derivatives...
  49. T

    Differentiable off and even functions

    Homework Statement Let f: R\rightarrow R be a differentiable even function. Prove that f' is an odd function. Also, prove that if f is a differentiable odd function, then f' is an even function. Homework Equations The Attempt at a Solution I tried to use definition, so I should...
  50. E

    Projection of a differentiable manifold onto a plane

    For a game I am thinking about making I would need to know how to project points from a differentiable bounded 3-manifold to a Euclidean plane (the computer screen). The manifold would be made from a 3-dimensional space with two balls cut out of it and a hypercylinder glued onto it at the holes...
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