Derivative Definition and 1000 Threads

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances.
The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable.
Derivatives can be generalized to functions of several real variables. In this generalization, the derivative is reinterpreted as a linear transformation whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables. It can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector.
The process of finding a derivative is called differentiation. The reverse process is called antidifferentiation. The fundamental theorem of calculus relates antidifferentiation with integration. Differentiation and integration constitute the two fundamental operations in single-variable calculus.

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  1. nomadreid

    [Solved] Unwanted minus sign in this derivative of a circular curve

    y'(t)/x'(t) = cos t/-sin t = -cot t. But as t is the angle, and the derivative is the slope, then isn't the slope supposed to be tan t?
  2. brotherbobby

    The derivative of the dot product "distributes" over each vector

    Drawing : I draw a diagram explaining the situation to the right. The vectors ##\mathbf{A}## and ##\mathbf{B}## are drawn at some time ##t## making an angle ##\theta## between them. At some later time ##t+dt##, the same vectors have changed to ##\mathbf{A}(t+dt)## and ##\mathbf{B}(t+dt)## while...
  3. S

    Derivative Problem: f’(1), f’(2), f’(3), and f’(5)

    1. How to get f’(1), f’(2), f’(3), and f’(5) 2. How to calculate average speed change of y to changes in x in the interval [0,6] 3. Estimate value of f’+(0) and f’-(6) pls help me about this graph, i dont know how to read this 🙏
  4. N

    B Does U-Substitution Function as the Inverse of the Chain Rule?

    Can someone please give as simple an example as possible to show what U substitution is about? I know basic integration rules but don't understand the point of u-substitution. I've read that it's used to "undo the chain rule", but I don't see how, and don't see how we can spot when we'd need to...
  5. M

    I Derivative w.r.t. time of relative position in special relativity

    In special relativity, is the derivative with respect to coordinate time of relative position equal to relative velocity? Does it matter if constant velocity is used?
  6. binbagsss

    I Question about partial derivative relations for complex numbers

    Apologies this is probably a very bad question but it's been a while since I have seen this. I have ##z=x+iy##. I need to convert ##\frac{\partial \psi(z)}{\partial z}## , with ##\psi## some function of ##z##, in terms of ##x## and ##y## I have ##dz=dx+idy##. so ##\frac{\partial \psi }{\partial...
  7. billtodd

    A A question from Superspace and 1001 lessons

    I believe that ##f'(\Phi(z))=\frac{df(\Phi(z))}{dz}##, I get confused with the prime in ##z'## and is it really just this derivative? I wonder how many people read this 1983 book.
  8. FQVBSina_Jesse

    A How to take the spatial derivative of quaternions

    I have a 5x5x5 set of grid points in space. I can describe each point with p(x,y,z), and I can convert them to spherical or other coordinates. At each point, I have a quaternion assigned to it. So, numerically, I can describe a q(x,y,z) quaternion field. The goal is to obtain a functional form...
  9. M

    Proving differentiability for a function from the definition

    For this problem, The solution is, However, does someone please know why we allowed to assume that the derivative exists for f i.e ##f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}##? Thanks!
  10. highschoolstudent454

    B Is the time derivate of force equal to the position derivative power?

    Is this wrong?
  11. D

    Is this a valid representation of a derivative?

    If ##f(g(x)) =f(x+h)##, then ## \lim_{h\to 0} \frac{f(g(x))-f(x)}{h}= \lim_{h\to 0} \frac{f(x+h)-f(x)}{h}=f'(x) ##. What if we let ##g(x) = x+\sin(h)##? Then ## \lim_{h\to 0} \frac{f(g(x))-f(x)}{h}= \lim_{h\to 0} \frac{f(x+\sin(h))-f(x)}{h}## This is not equivalent to that the standard...
  12. Safinaz

    I Partial derivative is terms of Kronecker delta and the Laplacian operator

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  13. G

    I How to apply tensor transformation rule

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  14. H

    I Intuition regarding Riemann curvature tensor

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  15. S

    Proof related to derivative and Big O notation

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  16. T

    Confusion on product rule for mass of differential volume element

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  17. Hak

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  18. abobik

    Find the partial diameter error of the surface area of cylinder

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  19. L

    What is the derivative of ln(x)^e ?

    Can't figure it out, here's a screenshot with better typography.
  20. M

    I Finding the derivative of a characteristic function

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  21. chwala

    Find the derivative of the given function

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  22. mcastillo356

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  23. nomadreid

    I Want to understand how to express the derivative as a matrix

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  24. N

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  25. Kumail Haider

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  26. Baela

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  27. Baela

    A Covariant derivative of Weyl spinor

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  28. snoopies622

    B Easy derivative but with a pesky singularity

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  29. M

    Computing the derivative of an exponential function

  30. binbagsss

    I 4d integration/differentiation notation and the total derivative

    This is probably a stupid question but, ## \frac{d\partial_p}{d\partial_c}=\delta^p_c ## For the notation of a 4D integral it is ##d^4x=dx^{\nu}##, so if I consider a total derivative: ##\int\limits^{x_f}_{x_i} \partial_{\mu} (\phi) d^4 x = \phi \mid^{x_f}_{x_i} ## why is there no...
  31. homeworkhelpls

    I Solving an equation with a parameter and a derivative

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  32. C

    The derivative of uv wrt x using st function (homework problem)

    TL;DR Summary: I attempt to find the derivative of uv with respect to x using non standard analysis, hyperreals, and the standard part function st; I take u to be a function of x, and I also take v to be a function of x. Hello everyone! I've been learning about non standard analysis concepts...
  33. iceninja3

    Finding acceleration from Velocity vs Position graph

    The answer is E. I was initially very confused as to why the answer was not A but realized that the graph was velocity vs position (rather than velocity vs time) which means I can't simply take the derivative of the given graph. One thing I tried was writing out the equation first(c being a...
  34. J

    Chemistry Why is ethanol considered the parent chain in naming this benzene derivative?

    The name of this molecule is 1‐(3‐nitrophenyl)ethanol. I'm confused why ethanol is treated as the parent chain in this case, not the phenyl group. If the ring is composed of more atoms, should it be the parent chain? Thank you.
  35. AJSayad

    A Thermodynamics Derivative Reduction Problems

    Hi everyone, I'm in a graduate level mechanical engineering thermodynamics class. We're working on derivative reductions using the gibbs and maxwell relations. I was wondering if anyone has any good sources of practice problems that I could use. I've looked through my textbook and there are...
  36. N

    I Finding Max and Min Extremes of a Function with Second Derivatives Equal to Zero

    What should I do when the f(x, y) function's second derivatives or Δ=AC-B² is zero? When the function is f(x) then we can differentiate it until it won't be a zero, but if z = some x and y then can I just continue this process to find what max and min (extremes) it has? What I've done is...
  37. S

    Understanding how time derivative = acceleration

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  38. N

    How do I calculate physics formulas containing derivatives and real numbers?

    Hi, I'm trying to calculate my own physics problem but didn't get it something. When I'm trying to calculate the impulse of the object when it's hit by F=10N force in the smallest possible time, then should I write: dP/dt = Fnet => dP = Fnet*dt ? Another question: In general, if I calculate...
  39. Bishal Banjara

    I Covariant Derivative Rank 2 Contravariant Tensor

  40. Delerion24

    Help with the derivative of the Dirac delta

    My goal is to develop the equation 21. You should asume that \delta(r_2-r_1)^2 =0. These is named renormalization. Then my question is , do my computes are correct with previous condition ?
  41. Fady Megally

    I Second derivative, chain rules and order of operations

    So the chain rule for second derivatives is $$ \frac {d^2 y} {d t^2} = \frac{d}{dx}(\frac {dy} {dx}) \cdot \frac {dx} {dt} \cdot \frac {dx} {dt} + \frac {dy} {dx} \cdot \frac {d^2 x} {d t^2} = \frac{d^2 y}{d x^2} \cdot (\frac {dx} {dt})^2 + \frac {dy} {dx} \cdot \frac {d^2 x} {d t^2}$$ Today I...
  42. H

    Calculating total derivative of multivariable function

    This isn't a homework problem exactly but my attempt to derive a result given in a textbook for myself. Below is my attempt at a solution, typed up elsewhere with nice formatting so didn't want to redo it all. Direct image link here. Would greatly appreciate if anyone has any pointers.
  43. L

    B Question about the definition of a partial derivative

    I just started to study thermodynamics and very often I see formulas like this: $$ \left( \frac {\partial V} {\partial T} \right)_P $$ explanation of this formula is something similar to: partial derivative of ##V## with respect to ##T## while ##P## is constant. But as far as I remember...
  44. P

    I Definition of functional derivative

    In the book Quantum Field Theory for the Gifted Amateur, they define the functional derivative as: $$ \frac{\delta F}{\delta f(x))} = \lim_{\epsilon\to 0} \frac{F[f(x') + \delta(x'-x)) ] - F[ f(x') ]}{\epsilon} $$ Why do they use the delta function and not some other arbitrary function?
  45. Vladimir_Kitanov

    Derivative of ##\vec s = \vec \theta \times \vec r##

    My try: ##\vec s = \vec \theta \times \vec r## ##\frac {d}{dt} (\vec s) = \frac {d}{dt} (\vec \theta \times \vec r)## ##\vec v = \frac {d}{dt}(\vec \theta) \times \vec r + \vec \theta \times \frac {d}{dt}(\vec r)## ##\vec v = \vec \omega \times \vec r + \vec \theta \times \vec v## And I...
  46. G

    I Understanding Covariant and Partial Derivatives in General Relativity

    In the 128 pages of 《A First Course in General Relativity - 2nd Edition》:"The covariant derivative differs from the partial derivative with respect to the coordinates only because the basis vectors change."Could someone give me some examples？I don't quite understand it.Tanks！
  47. R

    I Inequality with integral and max of derivative

    Hi. I was reading Lighthill, Introduction to Fourier Analysis and Generalised Functions and in page 17 there is an example/proof where I can't make sense of the following step: $$ \left| \int_{-\infty}^{+\infty} f_n(x)(g(x)-g(0)) \, \mathrm{d}x \right| \le \max{ \left| g'(x) \right| }...
  48. chwala

    Find the second derivative of the relation; ##x^2+y^4=10##

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  49. brotherbobby

    Spirit evaporating from a bowl

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  50. N

    The connection between potential energy and force

    Hi, if the force is the derivative of potential energy, does it mean that the force is equal to mg and with a constant gravity, it will be the same at any height? But in real life, F (or mg) would be different on the Earth's surface and 400 km above it (~8 m/s^2). So, this formula is used to...
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