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. R

    I Is the Jacobian Directional Derivative for a Sphere Correct?

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  2. D

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  3. karush

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  4. beyondlight

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  5. N

    Is This Calculation of ∂z/∂x Correct for the Given Function?

    Homework Statement ∂z/∂x of ycos(xz)+(4xy)-2z^2x^3=5x[/B] Homework Equations n/a The Attempt at a Solution ∂z/∂x=(5+yz-4y+6z^2x^2)/(-yxsin(xz)-4zx^3)[/B] Is this correct? Just trying to make sure that's the correct answer. I appreciate the help. I can post my work if need be. Thanks
  6. Euler2718

    I General Nth Derivative f(x)=x^x: Solving a Difficult Problem

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  7. mnb96

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  8. N

    Newtons Divided Difference First Derivative

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  9. Jonathan Densil

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  10. Z

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  11. kevin2016

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  12. Destroxia

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  13. B

    2nd derivative change of variables

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  14. ZKhawla

    Dirac derivative and signal energy distribution

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  15. G

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  16. G

    Finding the total distance traveled by the body at interval

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  17. O

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  18. B

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  19. heartshapedbox

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  20. Y

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  21. D

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  22. C

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  23. BreCheese

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  24. Drakkith

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  25. PhysicsKid0123

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  26. Rmehtany

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  27. E

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    Hello, I have this problem \frac{\partial}{\partial\,x}\int_0^{∞}\log(1+x)\,f_X(x)\,dx, where x is a random variable, and f_X(x) is its probability density function. It's been a long time since I encountered a similar problem, and I forgot how to do this. Do we use Leibniz integral rule...
  28. Q

    Tannor Quantum Mechanics derivative of variance of position

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  29. T

    What is the (higher order) time derivative of centripetal acceleration?

    Just using basic dimensional analysis, it appears the time derivative of centripetal acceleration is ## \vec{r} \omega^3 ##, but this intuitive guess would also extend to higher order time derivatives, no? Implying: ## \frac {d^n \vec{r}}{dt^n} = \vec{r} \omega^n ## It seems to follow from the...
  30. M

    Can Complex Derivatives Clarify Div and Curl Properties?

    In trying to get an intuition for curl and divergence, I've understood that in the case of R2, div f(x,y) = 2Re( d/dz f(z,z_)) and curl f(x,y) = 2Im( d/dz f(z,z_)), where f(z,z_) is just f(x,y) expressed in z and z conjugate (z_). Is there any way of proving the fundamental properties of div and...
  31. M

    Is ln(x) differentiable at negative x-axis

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  32. J

    Implications of varying the definition of the derivative?

    I have been playing around with calculus for a while and I wondered what would it be like to make some changes to the definition of derivatives. I'd like to look at the original definition of derivatives in this way (everything is in lim Δx→0): F(x+Δx) - F(x) = F'(x) * Δx The Δx factor...
  33. ognik

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  34. O

    Derivative of Dot Product via Product Rule, commutative?

    Homework Statement Basically, I'm looking at the property that says if the magnitude of a vector valued function is constant, then the vector function dotted with it's derivative will be zero. But I'm stuck towards the end because the proof I found online seems to skip a step that I'm not...
  35. 5P@N

    Is the derivative of 2x^2 = 4x or 8x?

    Call me crazy, but I do recall the power rule of integration viz: f(x) = x^n, f(x)' = n*x^n-1. Therefore, it seems as though 2x^2 would have a derivative of 4x. Fine. So why have I encountered someone else claiming that it's 8x? WHAT?! Who's right?
  36. T

    Def. of derivative and cosx=sin(Pi/2-x) to prove y'=-sinx

    A lot of web pages/books show how to use cosx=sin(Pi/2-x) and the chain rule to prove that the derivative of cosx=-sinx. My question is how to use this identity and the defintion of the derivative to prove the same thing. Or whether it is at all possible. Seeing that i get...
  37. avito009

    What is the derivative of the following?

    In the book that I am reading the derivative of y = bpx is bpxxlogep. How?
  38. I

    Can I undo a derivative to solve for a variable?

    If I have an equation where there is a derivative surrounding the variable, how do I undo the derivative and solve for the variable? Example would be- A= dx/dy when x=m*v*λ-2 and y=y Solving for v. I am a beginner so please explain thoroughly.
  39. P

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  40. FrancescoS

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  41. REVIANNA

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  42. lucphysics

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  43. S

    Partial derivative of a complex number

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  44. R

    Multivariable partial derivative

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  45. H

    Must the 1st derivative of phi be undetermined at V=infinity

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  46. karush

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  47. B

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  48. R

    MHB What is the 2nd Derivative of y(t)=tan5t?

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  49. B

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  50. C

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