# Derivative Definition and 143 Discussions

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. ### I Must functions really have interval domains for derivatives?

Nearly every analysis reference I come across defines the derivative for functions on an open interval ##f:(a, b) \rightarrow \mathbb{R}##. I understand that, in constructing the definition of ##f## being differentiable on a point ##c##, we of course want it to first be a point it's domain, so...
2. ### Find the equation of a tangent line to y = x^2?

Homework Statement the line goes through (0, 3/2) and is orthogonal to a tangent line to the part of parabola y = x^2, x > 0 Homework Equations The Attempt at a Solution I have problems regarding finding the equation of tangent line to the part of parabola because the question not...
3. V

### Calculus in the velocity and acceleration of satellites.

Homework Statement I am working on a project dealing with the velocity and acceleration of satellites based on their distance from Earth. I was recommended to include some calculus in this project. Originally I thought I could just take the derivative of the orbital speed equation to find...
4. ### I don't understand the derivation process here, help?

Homework Statement I understand the derivation it showed that included the sin (15.7 in the image) I just don't understand the following (15.8 in the image). Does "t" get pulled out of the equation? If so what do we derive for then? Does it become 0? If so, it would remain 0 and sin(0) is just...
5. ### I Derivative of 4^x

On my exam, we had to find the derivative of 4^x. This is what I did Y=4^x lny=xln4 y=e^xln4 and then finding the derivative for that I got, (xe^(xln4))/4 My professor said that it was wrong and even after I told her what I did to get the answer. She told me the answer was (4^x)ln4 . Which I...
6. ### Laplace transform of derivative of convolution

Prelude Consider the convolution h(t) of two function f(t) and g(t): $$h(t) = f(t) \ast g(t)=\int_0^t f(t-\tau) g(\tau) d \tau$$ then we know that by the properties of convolution $$\frac{d h(t)}{d t} = \frac{d f(t)}{d t} \ast g(t) = f(t) \ast \frac{d g(t)}{d t}$$ Intermezzo We also know that...
7. ### How to solve this partial derivative which includes a summation?

I was reading a research paper, and I got stuck at this partial differentiation. Please check the image which I have uploaded. Now, I got stuck at Equation (13). How partial derivative was done, where does summation gone? Is it ok to do derivative wrt Pi where summation also includes Pi...
8. ### Problem with a derivative

Hello PF, 1. Homework Statement I've been having problems with the deriative of a function, although I thought I've done everything right, my solution doesn't match with the right solution. I have no clue what (or if) I've done anything wrong, or simply don't know the tricks I was supposed to...
9. ### Insights The Pantheon of Derivatives - Part II - Comments

fresh_42 submitted a new PF Insights post The Pantheon of Derivatives - Part II Continue reading the Original PF Insights Post.
10. ### Insights The Pantheon of Derivatives - Part I - Comments

fresh_42 submitted a new PF Insights post The Pantheon of Derivatives - Part I Continue reading the Original PF Insights Post.
11. ### How to find absolute min/max of f(x)=x^3+3x^2-24x+1 on[-1,4]

Homework Statement How to find absolute min/max of f(x)=x^3+3x^2-24x+1 on[-1,4] I need to find the absolute min and absolute max. Homework Equations The Attempt at a Solution I first took the derivative, reduced, and set it to equal 0 to find crit numbers. x^2+2x-8=0 Factored. (x+4)(x-2) =...
12. ### Minimum and Maximum Values

Homework Statement The problem is in the attached file. The part I need a little help with is part b. Homework Equations and attempt at a solution[/B] For part a, I got h(8) = 2, h'(6) = -2, and h''(4) = -2. For part c, I found that the integral from 0 to 5 is 7, so I multiplied 7 by 7 to...
13. ### Finding Second and Third Derivative from Graph

Homework Statement The problem asks to find g'(2), g''(2), and g'''(4). Homework Equations and attempt at solution[/B] The derivative of g(x) is just the function f(x). So g'(2) = f(2) = -2. I'm not sure how to find g''(2) and g'''(4). I understand that g''(2) is f'(2), but how do I find...
14. ### Differentiation question

Homework Statement [/B] Summarizing: two civilizations hate each other, one of the civilizations throws a curse at the second. The second civilization succumbs to chaos and has a change in Population each week of ΔP= -√P. That is: Pn = Pn-1-√Pn-1 Homework Equations [/B] Considering that the...
15. ### Finding the Electric Field given the potential in spherical

Homework Statement The problem statement is in the attachment Homework Equations E[/B] = -∇φ ∇ = (∂φ/∂r)er The Attempt at a Solution I am confused about how to do the derivative apparently because the way I do it gives E = - (∂[p*r/4πε0r3]/∂r)er = 3*(p*r)/4πε0r4er
16. ### Kinematics, deriving equations.

Homework Statement "Derive the equations for position (in terms of acceleration, initial position, initial velocity, and time) and velocity (in terms of constant acceleration, a, initial velocity, v0, and time, t) from the definitions of position, velocity, and acceleration (derivative...
17. ### Derivative Question

Homework Statement [/B] I've tried to search this up but to no avail. How am I suppose to solve this: d2y/dx3 Homework Equations N/A The Attempt at a Solution Here's what I think I need to do: 1: Square and cube y and x respectively. 2: Find the second and third derivative of y and x...

27. ### I Need help with derivative notation

If I have a scalar function of a variable ##x## I can write the derivative as: ##f'(x)=\frac{df}{dx}##. Now suppose ##x## is no longer a single variable but a vector: ## x=(x^1, x^2, ..., x^n)##. Then of course we have for the derivative ##(\frac{\partial f}{\partial x^1}, ..., \frac{\partial...
28. ### A Create function which meets slope and point requirements

I am trying to create a function of A and x which has the following properties. A is a scaling parameter that determines the shape of the function. I write the function below in f(A,x) form 1) f(A,1)=1 always 2) For all x>1, 0<f ' (x)<1 3) As A approaches some upper bound (which could be...
29. ### A Relationship between metric tensor and position vector

Given the definition of the covariant basis (##Z_{i}##) as follows: $$Z_{i} = \frac{\delta \textbf{R}}{\delta Z^{i}}$$ Then, the derivative of the covariant basis is as follows: $$\frac{\delta Z_{i}}{\delta Z^{j}} = \frac{\delta^2 \textbf{R}}{\delta Z^{i} \delta Z^{j}}$$ Which is also equal...
30. ### I Substitution in a Lebesgue integral

Hi, friends! I read that, if ##f\in L^1[c,d]## is a Lebesgue summable function on ##[a,b]## and ##g:[a,b]\to[c,d]## is a differomorphism (would it be enough for ##g## to be invertible and such that ##g\in C^1[a,b]## and ##g^{-1}\in C^1[a,b]##, then...
31. ### Need a little help with this related rates problem

Can someone help me with this? (dA/dt)=1cm/s (cm^2 whatever...leave out trivial corrections). A=pir^2 (dA/dt)=2pir(dR/dt) Multiply through by (1/2pir) (dA/dt)/(2pir)=dR/Dt What is the rate of change of the radius for a circumfrance of 2 I just used the related rates formula that I derived for...
32. ### I I can't understand the fact

I can's understand the fact about the equation i can't prove the equation from the first attachment to the second attachment pls help. Sorry for bad english
33. ### I How to Convert this Derivative?

I have a derivative of a function with respect to ##\log \left(r\right)##: \begin{equation*} \frac{dN\left(r\right)}{d \log\left(r\right)} = \frac{N}{\sqrt{2\pi} \log\left(\sigma\right)} \exp\left\{-\frac{\left[\log \left(r\right) - \log\left(r_M\right)\right]^2}{2...
34. ### I What is the derivative of a matrix transpose?

Hi! As the title says, what is the derivative of a matrix transpose? I am attempting to take the derivative of \dot{q} and \dot{p} with respect to p and q (on each one). Any advice?
35. ### Ladder operator commutator with arbitary function

Hey there! 1. Homework Statement I've been given the operators a=\sqrt\frac{mw}{2\hbar}x+i\frac{p}{\sqrt{2m\hbar w}} and a^\dagger=\sqrt\frac{mw}{2\hbar}x-i\frac{p}{\sqrt{2m\hbar w}} without the constants and definition of the momentum operator: a=x+\partial_x and a^\dagger=x-\partial_x with...
36. ### I How to understand the notion of a limit of a function

I am trying to explain to someone the formal notion of a limit of a function, however it has made me realize that I might have some faults in my own understanding. I will write down how I understand the subject and would very much appreciate if someone(s) can point out any...
37. ### IalChange of variables/verifying solution

Homework Statement Trying to use change of variables to simplify the schrodinger equation. I'm clearly going wrong somewhere, but can't see where. Homework Equations [/B] Radial Schrodinger: -((hbar)2)/2M * [(1/r)(rψ)'' - l(l+1)/(r^2) ψ] - α(hbar)c/r ψ = Eψ The Attempt at a Solution...
38. ### I Magnitude of the Second Derivative

So to find the x values of the stationary points on the curve: f(x)=x3+3x2 you make f '(x)=0 so: 3x2+6x=0 x=0 or x=-2 Then to find which of these points are maximum or minimum you do f ''(0) and f ''(-2) so: 6(0)+6=6 6(-2)+6=-6 so the maximum has an x value of -2 and the minimum has an x value...
39. ### Uncertainty of Derivatives

Homework Statement I know this is more of a physics question, but I tried there and wasn't successful. I have done a physics experiment measuring the weight as a function time of the discharge of water from a cylindrical bottle with a pinhole at the bottom. What I ultimately want to get at is...
40. ### A What is the closed-form solution using ALS algorithm to optimize

C \in \mathbb{R}^{m \times n}, X \in \mathbb{R}^{m \times n}, W \in \mathbb{R}^{m \times k}, H \in \mathbb{R}^{n \times k}, S \in \mathbb{R}^{m \times m}, P \in \mathbb{R}^{n \times n} ##{S}## and ##{P}## are similarity matrices (symmetric). ##\lambda##, ##\alpha## and ##\beta## are...
41. ### Dirac derivative and signal energy distribution

Hi, I'm writing a mathematical expression of energy distribution of a signal, and in the formula I’ve found first and second derivative of delta function. I have to analyze my result but couldn’t found how to read these two derivative from an energy point of view. And how can we see further...
42. ### Find the derivative of the function(Quotient rule)

Homework Statement Find the derivative of the function y = (3-2x^3+x^6 )/x^9 Homework Equations Derivatives The Attempt at a Solution I have tried to use the quotient rule and got to -6x^11 + 6x^14 - 27x^8 + 18x ^24 - 9x ^14 / (x^9)^2 Which doesn't look close to the answer -27/x^10 +...
43. ### Finding the total distance traveled by the body at interval

Homework Statement At time t, the position of a body moving along the s-axis is s= t^3 -12t^2 + 36t m(meters) Find the total distance traveled by the body from t = 0 to t = 3. Homework Equations Derivatives The Attempt at a Solution I got the derivative which is 3t^2 - 24t + 36(meters) I...
44. ### A nasty integral to compute

Hey Guys! I was working on an integration problem, and I "simplified" the integral to the following: $$\int \limits_0^{2\pi} (7.625+.275 \cos(4x))^{1.5} \cdot (A \cos(Nx) + B \sin(Nx)) \cdot (Z-v \cos(x)) dx$$ This integral may seem impossible (I have almost lost all hope on doing this...
45. ### Is ln(x) differentiable at negative x-axis

Since lnx is defined for positive x only shouldn't the derivative of lnx be 1/x, where x is positive. My books does not specify that x must be positive, so is lnx differentiable for all x?
46. ### 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...
47. ### 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...
48. P

### Snell's law demonstration?

I can't figure out why my demonstration of snell's law fails, that's the demonstration: (I used a photo) I think it fails because the function t (HO) represents a line and so the concept of minimum is not defined, when I take the derivative and equal it to 0 I'm considering the case when the...
49. ### Application of derivatives

Homework Statement the original function is ##−6 x^3−3x−2 cosx## ##f′(x)=−2x^2−3+2sin(x)## ##−2x^2 ≤ 0## for all x and ##−3+2 sin(x) ≤ −3+2 = −1##, for all x ⇒ f′(x) ≤ −1 < 0 for all x The Attempt at a Solution this problem is part of a larger problem which says there is a cubic...
50. ### Find stationary points of a two variable function involving

Homework Statement Find all stationary points of the function G(x, y) = (x^3)*e^(−x^2−y^2) Homework Equations fx=0 and fy=0 The Attempt at a Solution Gx = 3x^2*e^(-x^2-y^2) +x^3(-2x)e^(-x^2-y^2) = e^(-x^2-y^2)(3x^2-2x^4) Gx = 0 implies 3x^2-2x^4=0 x^2(3-2x^2)=0 hence x =0 ,+or-...