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. Q

    Lie Derivative of one-form: an identity

    Homework Statement I am trying to prove an identity for the Lie derivative of a smooth one-form. The identity is: for X, Y smooth vector fields, alpha a smooth one-form, we have: $$L_{[X, Y]}\alpha = [L_X, L_Y]\alpha$$ For anyone familiar with the book, this is exercise 5.26 in the first...
  2. T

    Derivative of an inverse function

    Homework Statement I will post a picture of the problem and then the second picture will be my work. The problems are the first two. Homework EquationsThe Attempt at a Solution I didn't know how to do this at first so I don't know if I am doing it correctly now. Also I don't know the correct...
  3. T

    Derivative of Natural Log: How to Solve Number 3 on Homework Assignment

    Homework Statement I posted a picture of it and my attempt it is number 3 Homework EquationsThe Attempt at a Solution I tried using log properties and I am not sure what went wrong and how to arrive at the correct answer. Mod note: Messy, disorganized image deleted.
  4. S

    New Calc student w/ a derivative question

    Homework Statement Hello all, thank you for the help in advance. It's a two-sided derivative problem, for lack of a better term, and I appreciate all hints or help. If we have a function y so that y=bx for all x<0, and y= x^2-13x for all x> or = 0, for what value of b is y differentiable at...
  5. kostoglotov

    Q about 2nd derivative test for multivariable functions

    Homework Statement So the test is to take the determinant (D) of the Hessian matrix of your multivar function. Then if D>0 & fxx>0 it's a min point, if D>0 & fxx<0 it's a max point. For D<0 it's a saddle point, and D=0 gives no information. My question is, what happens if fxx=0? Is that...
  6. M

    What is the difference between curly and derivative (d) sign

    Dear All, Please see the image below in attachment where Energy is function of K. I want to understand how is it possible to understand the last expression ( dE = ? ). Additionally, what is the difference between curly and derivative (d) sign ? Many thanks to the mentors on this forum Best wishes
  7. Drakkith

    Simplifying a Function Prior to Finding a Derivative

    Okay guys, this is driving me absolutely nuts. I'm working on finding derivatives using the product and quotient rules and the book will sometimes simplify the problem before finding the derivative but sometimes wont and I don't understand why. For example: The function y = (v3-2v√v)/v The book...
  8. N

    Taking the time derivative of a curl

    Is the time derivative of a curl commutative? I think I may have answered this question... Only the partial time derivative of a curl is commutative? The total time derivative is not, since for example in cartesian coordinates, x,y,and z can themselves be functions of time. In spherical and...
  9. M

    Chain Rule with Partial Derivative?

    Homework Statement Given that the surface x^7y^2+y^4z^6+z^8x^8+9xyz=12 has the equation z=f(xy) in a neighbourhod of the point (1,1,1) with f(x,y) differentiable, find the derivatives. df/dx (1,1) = ? d^2f/dx^2 (1,1) = ? Homework EquationsThe Attempt at a Solution df/dx (1,1) I got -24/23 or...
  10. kostoglotov

    Partial Derivative Q: continuity and directional deriv's

    Homework Statement a) Show that the function f(x,y)=\sqrt[3]{xy} is continuous and the partial derivatives f_x and f_y exist at the origin but the directional derivatives in all other directions do not exist b) Graph f near the origin and comment on how the graph confirms part (a). 2. The...
  11. T

    Material Derivative (Convective Derivative Operator)

    Hi, I've learned that material derivative is equal to local derivative + convective derivative, but can't seem to find out which way the convective derivative acts, like for example in velocity fields: The equation my teacher gave us was (with a and v all/both vectors): Acceleration = material...
  12. C

    Kinetic Energy Time Derivative

    Homework Statement So the first part asks to prove the time derivative of kinetic energy is dT/dt=F dot product v which I did not problem. but then the second part of the problem asks to prove that if the mass is changing with time then the time derivative of d(mT)/dt=F dot product m and I'm...
  13. powerof

    Symmetry in second order partial derivatives and chain rule

    When can I do the following where ##h_{i}## is a function of ##(x_{1},...,x_{n})##? \frac{\partial}{\partial x_{k}}\frac{\partial f(h_{1},...,h_{n})}{\partial h_{m}}\overset{?}{=}\frac{\partial}{\partial h_{m}}\frac{\partial f(h_{1},...,h_{n})}{\partial x_{m}}\overset{\underbrace{chain\...
  14. V

    Multivariate piecewise fxn continuity and partial derivative

    1. Problem Define a function: for t>=0, f(x,t) = { x for 0 <= x <= sqrt(t), -x + 2sqrt(t) for sqrt(t) <= x <= 2sqrt(t), 0 elsewhere} for t<0 f(x,t) = - f(x,|t|) Show that f is continuous in R^2. Show that f_t (x, 0) = 0 for all x. Then define g(t) = integral[f(x,t)dx] from -1 to 1. Show...
  15. T

    MHB Derivative of secx where x = pi/3

    The derivative of secx is $$\d{y}{x} secx =secx tanx $$ But if $$x = \frac{\pi}{3}$$, then $$secx = 2 $$ and the derivative of a constant is 0. And $$sec\frac{\pi}{3} tan\frac{\pi}{3}$$ is equal to $$\frac{3}{2}$$ So what is the derivative of $$secx$$ where $$x = \frac{\pi}{3}$$?
  16. C

    Index-Free Decomposition of Derivative Timelike Congruence

    Suppose we have a general timelike congruence of curves with tangent vector field ##V##, then the standard decomposition of the covariant derivative in index form (see e.g. Hawking and Ellis' "Large scale structure of space and time" equation 4.17) is given by $$V_{a;b} = \omega_{ab} +...
  17. stevendaryl

    Derivative of Vector Field: Connection Needed?

    Some context for my question: If you have a smooth manifold \mathcal{M} you can define tangent vectors to parametrized paths in the following way: If \mathcal{P}(s) is a parametrized path, then \frac{d}{ds} \mathcal{P}(s) = V where V is the differential operator that acts on scalar fields...
  18. H

    Derivative of the Product of Two Functions: Applying the Chain Rule

    Homework Statement take the derivative of a(t) = b(t)c(t) Homework Equations chain rule The Attempt at a Solution Apply the chain rule: a'(t) = c(t)b'(t) + b(t)c'(t) Is this correct? Thank you.
  19. goonking

    Solve Derivative of y = x2sinx

    Homework Statement y = x 2sinx Homework EquationsThe Attempt at a Solution Ok, so If I see an x in an exponent, I would want to use ln to 'bring it out', right? ln y = ln (x 2sinx) = ln x + ln 2sinx = ln x + sinx ln2 now I take the derivative : y'/y = 1/x + cosx ln2 multiply both sides...
  20. goonking

    Computing derivative (basic calculus question)

    Homework Statement Compute Derivative y = xx + sin(x) Homework EquationsThe Attempt at a Solution since I have x in the exponent (x^x), I multiply both sides by ln: ln y = ln xx + ln sin(x) the x in the exponent comes out into the front, right? y'/y = x ln x + ln sin (x) using product...
  21. L

    MHB Solve Derivative Questions with Quotient Rule

    Hello, I am struggling with these two questions. I think here should be used a quotient rule, but I am not sure how to proceed. a) f(x)=sin$\frac{1}{x}$ b)g(x)=$\frac{1}{sinx}$ Can someone please help. Thanks
  22. J

    Stuck getting derivative when can't isolate my variable

    Hi. This is not a homework assignment. I am working to get an extrema on a graph that involves a bunch of functions and got stuck on one step: How to get the derivative of: \frac{dy}{dn} = \frac{nc(a+b)}{nc+a} I can't get "n" in a place where I recognize how to get the derivative of it. I...
  23. ElijahRockers

    Nth derivative Fourier transform property

    Homework Statement I am given f(t) = e^-|t| and I found that F(w) = ##\sqrt{\frac{2}{\pi}}\frac{1}{w^2 + 1}## The question says to use the nth derivative property of the Fourier transform to find the Fourier transform of sgn(t)f(t), and gives a hint: "take the derivative of e^-|t|" I also...
  24. Antonija

    Comp Sci Derivative of a function in FORTRAN

    [Note from mentor: This thread was originally posted in a non-homework forum, therefore it does not follow the standard homework template.] ------------------------------------------------ Hello. I have some homework to do. I need to make program that finds minimum/maximum of a function od 2...
  25. D

    Proving the fundamental theorem of calculus using limits

    Would it be a legitimate (valid) proof to use an \epsilon-\delta limit approach to prove the fundamental theorem of calculus? i.e. as the FTC states that if f is a continuous function on [a,b], then we can define a function F: [a,b]\rightarrow\mathbb{R} such that F(x)=\int_{a}^{x}f(t)dt Then F...
  26. I

    What Directions at Point (2, 0) Make the Rate of Change -1 for f(x, y) = xy?

    Homework Statement In what directions at the point (2, 0) does the function f(x, y) = xy have rate of change -1?D_{u}(f)(a,b) = \bigtriangledown f(a,b)\cdot (u_{1}, u_{2}) f(x,y) = xy (a,b) = (2,0). The Attempt at a Solution \frac{\partial f}{\partial x} = y \frac{\partial f}{\partial y} =...
  27. Calpalned

    Partial Derivative of w = xe^(y/z) | Homework Solution

    Homework Statement Find the partial derivative of ## w = xe^\frac {y}{z} ##. Homework Equations N/A The Attempt at a Solution ## \frac{∂f}{∂x} = e^y/z ## ## \frac{∂f}{∂y} = \frac{xe^y/z}{z} ## ## \frac{∂f}{∂z} = (-yz^-2)(xe^yz^-1) ## Are theses correct? Thanks everyone.
  28. P

    Components of vector derivative

    In the book "Introduction to Mechanics" by K&K, an increment of a generic time-varying vector is split into two components, ##\Delta \vec{A} _{\perp}## and ##\Delta \vec{A}_{\parallel}##. Their magnitudes are approximated by: $$A \Delta \theta$$ and $$\Delta A$$ respectively. (Where ##\Delta...
  29. I

    How Do You Find the Gradient Vector from a Directional Derivative?

    Homework Statement D_{u}(f)(a,b) = \triangledown f(a,b)\cdot u D_{(\frac{1}{\sqrt2}, \frac{1}{\sqrt2})}(f)(a,b) = 3 \sqrt{2} where u = (\frac{1}{\sqrt2}, \frac{1}{\sqrt2}) find \bigtriangledown f(a.b) Homework EquationsThe Attempt at a Solution first you change grad f into it's partial...
  30. D

    Parallel Transport & Covariant Derivative: Overview

    I have been reading section 3.1 of Wald's GR book in which he introduces the notion of a covariant derivative. As I understand, this is introduced as the (partial) derivative operators \partial_{a} are dependent on the coordinate system one chooses and thus not naturally associated with the...
  31. P

    Can the Time Derivative of a Vector Change if its Magnitude or Angle Decreases?

    In chapter 1 of the book "Introduction to Mechanics" by Kleppner and Kolenkow, the derivative of a generic vector ##\vec{A}## is discussed in terms of decomposing an increment in ##\vec{A}##, ##Δ\vec{A}##, into two perpendicular vector vectors; one parallel to ##\vec{A}## and the other...
  32. E

    Find two angles where the directional derivative is 1 at p0

    1. Given a function f(x,y) at (x0,y0). Find the two angles the directional derivative makes with the x-axis, where the directional derivative is 1. The angles lie in (-pi,pi]. 2. f(x,y) = sec(pi/14)*sqrt(x^2 + y^2) p0 = (6,6) 3. I use the relation D_u = grad(f) * u, where u is the...
  33. H

    Bulk Modulus and its derivative in a fcc lattice

    The bulk modulus B = - V (∂P/∂V). At constant temperature the pressure is given by P= -∂U/∂V, where U is the total energy. We can write B in terms of the energy per particle u = U/N and volume per particle v = V/N : B = v...
  34. A

    How to complete this derivative proof?

    Homework Statement Assume that $f(x)$ has two derivatives in $(0,2)$ and $0<a<b<a+b<2$. Prove that if $f(a)\ge f(a+b)$ and $f″(x)\le 0$ $\forall x \in (0, 2)$, then: $$\frac{af(a)+bf(b)}{a+b} \ge f(a+b) \tag 1$$ Homework Equations Below The Attempt at a Solution**MY PROOF:** If $(1)$ is...
  35. L

    MHB Simplifying the Product Rule for Derivatives

    Hello, I have this exercise that I can't get the right answer. I have to find derivative of g(x)= (4${x}^{2}$-2x+1)${e}^{x}$ So, what is did is g$^{\prime}$=(8x-2)${e}^{x}$+(4${x}^{2}$-2x+1)${e}^{x}$ My Prof said it is wrong... I am not sure if I have to multiply the brackets or what I did...
  36. A

    Interesting Derivative Proof Question

    Homework Statement I recently searched around SE, and found: http://math.stackexchange.com/questions/1142546/how-to-solve-this-derivative-of-f-proofHomework Equations Below The Attempt at a Solution The answer is interesting. "A function given that $$f(x)=f''(x)+f'(x)g(x)$$ could be an...
  37. A

    MHB How Does Positive Second Derivative Imply Function Convexity in Calculus?

    I recently searched around SE, and found: [How to solve this derivative of f proof][1] [1]: calculus - How to solve this derivative of f proof? - Mathematics Stack Exchange The answer is interesting. "A function given that $f(x)=f''(x)+f'(x)g(x)$ could be an exponential function, sine...
  38. Calpalned

    What Is the Correct Partial Derivative of 6xyz?

    Homework Statement Find (∂z/∂x) of 6xyz Homework Equations N/a The Attempt at a Solution The correct answer is 6xy(∂z/∂x) but I would like proof of it. I got something different when I tried taking the partial derivative. 6xyz = 6x(yz) = Multiplication rule for derivatives 6(∂x/∂x) +...
  39. S

    Wave second order derivative equation

    Whenever the second order derivative of any physical quantity is related to its second order space derivative a wave of some sort must travel in a medium, why this is so?
  40. Suraj M

    Did I Apply the Derivative Rule Incorrectly for x²?

    i just started with calculus. There was this question my teacher asked us d/dx (x²) = 2x ... eq 1 now we can write 2² = (2+2) 3² = (3+3+3) 4²=(4+4+4+4) . . . n² = (n+n+n+n+...)n times so here d/dx (x²) = d/dx (x+x+x+...)x times so ⇒ d/dx (x) +d/dx(x) +...(x times) = 1+1+1+...(x times) = x ⇒d/dx...
  41. P

    Partial derivative of a square root

    Hi, I'm using partial derivatives to calculate propagation of error. However, a bit rusty on my calculus. I'm trying to figure out the partial derivative with respect to L of the equation: 2pi*sqrt(L/g) (Yep, period of a pendulum). "g" is assumed to have no error. I know I can use the...
  42. LiHJ

    How Do I Solve 1st and 2nd Derivative Homework Problems?

    Homework Statement Dear Mentors PF Helpers, Here's my question: I see it from my textbook with it solutions copied down below. Wonder is there another way to do it. Thank you for your time.Homework Equations [/B]The Attempt at a Solution
  43. M

    Finding 2nd partial derivative

    I've attached an image to this post. It essentially shows the equation for the first partial derivative using chain rule, which makes sense. What I'm confused with is how the second partial derivative was formulated. It seems they've simply squared the first partial derivative to find the second...
  44. A

    Parity operator commutes with second derivative?

    How do I prove that the parity operator Af(x) = f(-x) commutes with the second derivative operator. I am tempted to write: A∂^2f(x)/∂x^2 = ∂^2f(-x)/∂(-x)^2 = ∂^2f(-x)/∂x^2 = ∂^2Af(x)/∂x^2 But that looks to be abuse of notation..
  45. kelvin490

    Differentiability implies continuous derivative?

    We know differentiability implies continuity, and in 2 independent variables cases both partial derivatives fx and fy must be continuous functions in order for the primary function f(x,y) to be defined as differentiable. However in the case of 1 independent variable, is it possible for a...
  46. caters

    Derivative of a Fraction: Simplifying with Polynomial Division?

    Homework Statement if $$y = \frac{2x^5-3x^3+x^2}{x^3}$$ then $$\frac{dy}{dx} =$$ Homework Equations if $$f(x) = x^n$$ then $$f'(x) = nx^{n-1}$$ The Attempt at a Solution $$\frac{2x^5-3x^3+x^2}{x^3} = \frac{2x^5}{x^3} - \frac{3x^3}{x^3} + \frac{x^2}{x^3}$$ $$ f'(\frac{2x^5-3x^3+x^2}{x^3}) =...
  47. F

    Understanding Covariant Derivative & Parallel Transport

    Hello, I try to apprehend the notion of covariant derivative. In order to undertsand better, here is a figure on which we are searching for express the difference \vec{V} = \vec{V}(M') - \vec{V}(M) : In order to evaluate this difference, we do a parallel transport of \vec{V}(M') at point...
  48. Dakarai

    Derivative of a Utility Function

    Homework Statement What is the MRS of the quasilinear utility function U(q1, q2) = u(q1) + q2 ? Homework Equations MRS = - dU1/dU2 The Attempt at a Solution [/B]dU2 is 1 but I am unsure how to approach taking the derivative of u(q1). I have tried the answer as -dU and -dU * dq1...
  49. A

    Derivative involving an unknown constant

    Homework Statement Determine the value(s) of k such that y=-5k is the equation of the tangent line on the graph of F(x) = -x^2 + 4kx + 1 Homework Equations n/a The Attempt at a Solution not sure where to start this problem; but i understand some fundamentals here. I believe that the tangent...
  50. H

    Help with inverse of derivative function

    f(x) = 3x^3 + 3x^2+ 2x + 1 ,a = 3 formal is Homework is due tonight and this is the only problem i can't solve Your suppose to 3= 3x^3 + 3x^2 + 2x + 1 , solve for xThe find the derivative of y= 3x^3 + 3x^2 + 2x + 1 , then plug x into that and put it under 1.
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