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. Darth Frodo

    F(x) is differentiable at x =1, find f(x)

    Homework Statement suppose a function f is differentiable at x = 1 and lim[h → 0] \frac{f(1 + h)}{h} = 5 Finf, f(1) AND f'(1) The Attempt at a Solution f(x) is differentiable at x = 1 f(x) is continious at x = 1 lim[x→1] f(x)= f(1) f'(x) = lim[h→0] \frac{f(1 + h) -...
  2. Petek

    Continuity of the derivative of a decreasing differentiable function

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  3. F

    CR Equations: Real & Imaginary Parts Satisfy Cont. & Diff.

    Complex differentiable <--> real and imaginary parts satisfy C-R eqns and are cont. Say we have a complex function f(z) we can break this into real and imaginary parts: f(z)=u(x,y)+iv(x,y)In my book I am told the following:(1) f complex differentiable at z0 in ℂ --> the Cauchy Reimann...
  4. D

    MHB Continuous periodic piecewise differentiable

    Suppose that $f(\theta)$ is a continuous periodic piecewise differentiable function. Prove that $f(\theta) = f(0) + \int_0^{\theta}g(t)dt$ for a piecewise continuous $g$. I just need a nudge in the right direction here.
  5. P

    Is the function defined, continuous and differentiable

    Homework Statement Graph the function defined by the following. B = {(r/r0)B0 for r ≤ r0 {(r0/r)B0 for r > r0 (a) Is B continuous at r = r0? yes no (b) Is B differentiable at r = r0? Homework Equations The Attempt at a Solution I'm not exactly sure what to do...
  6. T

    Use Taylor's theorem to show a function is differentiable at x=0

    Homework Statement Use Taylor's theorem to estimate |(ex)-x-1| for 0≤x≤1. Thus prove that if a>(1/2) then: f(x)=(1-|x|a)*(ex)a is differentiable at x=0 Homework Equations The Attempt at a Solution So |(ex)-x-1|=(x^2)/2+(x^3)/6+(x^4)/24... But I don't see how this helps, I...
  7. A

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    f'(z0)=\stackrel{lim}{x\rightarrow0} \frac{f(z0-z)-f(z0)}{z} Hi, I'm attempting to use the above equation to show where z0 is not differentiable at some point z0 for the equation f(z) = |z|2 I was wondering how I could go about doing this? I tried letting z0 = a + bi, and z = x + yi...
  8. A

    Infinitely differentiable functions

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  9. I

    Infinitely differentiable function

    This might sound like a stupid question. f(x) = \begin{cases} &e^{-\frac{1}{x^2}} &\text{if } x\neq 0 \\ & 0 &\text{if } x = 0 \end{cases} Is the reason f is infinitely differentiable at 0 because we keep differentiating 0 as a constant, or because, \lim_{x\rightarrow 0} f`(x) =...
  10. N

    Even Differentiable Functions and Linearization

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  11. C

    Prove that a fuction is continous and differentiable everywhere, but not at f'=0

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  12. M

    Suppose that F: Rn -> Rn is continuously differentiable everywhere

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  13. O

    Where is this function differentiable?

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  14. R

    Continously differentiable f: R^n -> R^m not 1-1?

    Continously differentiable f: R^n --> R^m not 1-1? My course is over with now, but I never could figure out this question. It's pretty much been haunting me ever since, and the internet has not given me a proof that convinces me. My problem is determining why: A continuously di...
  15. H

    Proof that a function composition is differentiable

    Homework Statement Let f:R->R, differentiable, f(1)=1 and f'(1)=2. Homework Equations Prove that g:R->R such that g(x)=f(x)Arctg(f(x)) is differentiable in x=1 and calculate g'(1)The Attempt at a Solution I would prove it saying that if a function is differentiable then the product and...
  16. K

    Show that g(x):=|f(x)| is differentiable at c

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  17. R

    Show where the functions is anlaytic and differentiable

    Homework Statement z→x3+ i(1 - y)3: Show where the functions is analytic and differentiable. Homework Equations The Attempt at a Solution For a function to be analytic cauchy-riemann equations must hold.. so ux = vy and uy = -vx Now f(z) = x3 + i(1 - y)3 is already in the...
  18. S

    Is x^0 Differentiable at x=0?

    If the derivative of x^n equals nx^(n-1), then the derivative of x or x^1 equals x^0, but 0^0 is undefined. Does that mean x is not differentiable at zero?
  19. S

    Evaluate where F(x) is differentiable

    Hi there, I cannot seem to figure this question out. Homework Statement Let f: [0,3] -> R be defined as follows x if 0≤x<1, f(X)= 1≤x<2 x if 2≤x≤3 obtain formulas for F(x) = for 0≤x≤3 and sketch the graphs of f(x) and F(x). Where is F(x) differentiable? Evaluate F(x) where...
  20. O

    MHB Why isn't the "l1"-norm differentiable?

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  21. D

    Linear Algebra. Proving differentiable functions are a vector space.

    Question: Show the set of all differentiable functions on (-infinity, +infinity) that satisfy f′ + 2f = 0 is a vector space. I started the problem by assuming that f and g are both differentiable functions that satisfy this vector space. Then I ran through the ten axioms of addition and...
  22. C

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  23. F

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  24. F

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  25. F

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  26. F

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  27. J

    Suppose that f is a differentiable real function in

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  28. E

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  29. A

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  30. J

    Is showing that a series of functions is differentiable as simple as it appears?

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  31. P

    Differentiable on interval implies monotonic on some neighborhood of every point

    If f is differentiable on [a,b] and f'(c)>0 for some a<c<b then does this imply that f is monotonically increasing on some neighborhood of c? My intuition says yes but I just can't figure out a way to prove it. (not homework). Because of the weierstrass function I'm pretty sure...
  32. P

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  33. J

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  34. 5

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  35. G

    Is the following function differentiable?

    I have: http://img12.imageshack.us/img12/6121/capturerhf.png Is the function differentiable in (0,2)? If so, find its Tangent Plane. So far I have We have (\nabla f)(0,2)=(f_x(0,2).f_y(0,2))=\ldots=(0,1) , so if f is differentiable at (0,2) the only possible differential is \lambda...
  36. K

    Prove that, if f(x) is differentiable at x =c , then f(x) is continous at x=c

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  37. I

    Is f(z)=x differentiable with respect to z?

    Homework Statement The only thing given is f(z)=x. However, I am under the assumption that z is a complex variable where z=x+iy. I'm also assuming that x is a real variable. In this example, I know that f(z)=x is not differentiable with respect to z because it does not satisfy the...
  38. N

    Given a piecewise, prove that it is continous and differentiable

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  39. N

    If the graph of a differentiable function is symmetric

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  40. P

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  41. E

    Is the function differentiable everywhere?

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  42. M

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  43. Y

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    Homework Statement Prove that f(x) is a smooth function (i.e. infinitely differentiable) Homework Equations ln(x) = \int^{x}_{1} 1/t dt f(x) = ln(x) The Attempt at a Solution I was thinking about using taylor series to prove ln(x) is smooth but I'm strictly told to NOT assume f(x) = ln(x)...
  44. N

    Given a piecewise, prove that it is continous and differentiable

    Homework Statement For f(x)= { sin(x)/x if x≠0 , 1 if x=0. (a) Show that f is continuous and differentiable for all x. (b) Show the derivative f'(x) is continous. Homework Equations The Attempt at a Solution I know that if f is differentiable it is continous, so I need to focus...
  45. 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...
  46. M

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  47. A

    Show that f Uniform Differentiable implies f' Uniform Continuous

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  48. F

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  49. C

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

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  50. 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...
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