Derivatives Definition and 1000 Threads

I'm in calc 1 and want to make sure I'm understanding the reason that we find derivatives. From what I understand, a derivative is simply an equation for the rate of change at any given point on the original function. Is that correct? And the tangent line at point (x,y) is obtained by using...

let's assume that f(t) is a real function on [a,b], n is a positive integer (n-1)th derivative of f is continuous on [a,b], (n)th derivative exists for all t in (a,b) 1. Then can we say that (n-2)th derivative of f is continuous on [a,b]? 2. (n-2)th derivative of f is defined on a and b?

Hello PF! It's been a while since I last posted here. I have come across a problem in my textbook, which asks me to find expressions for V as a function of T and P, starting from the coefficients of thermal expansion and compressibility. \alpha = \frac{1}{V} \left(\frac{\partial V}{\partial T}...

Stressed first year university student here, fresh out of high school. I took physics in both grade 11 and 12, and thought I had a pretty good grasp on it; that is until this week. Introduction to derivatives and integrals to get from x(t) to v(t) to a(t) and vice-versa. I have a pretty good...

1) The line 2x+9=3 meets curve xy+y+2=0 at the points P and Q. Calculate the gradient of the curve at P and Q 2)Given that y=(x^2)/(x-2), find a) (d^2)y/dx^2 in its simplest form b)ther range of value for which dy/dx and (d^2)y/dx^2 are positive. I can't figure out either of the sums...

Why is \frac{d}{dt}\left | r(t) \right | =\frac{1}{\left | r(t) \right |} r(t)\cdot r'(t) ? A hint is given to me, saying \left | r(t)^2 \right | = r(t)\cdot r(t) . I think it's something to do with differentiating both side of the equation given as 'hint', but I have no idea how to execute...

The wikipedia article on \sgn (x) (http://en.wikipedia.org/wiki/Sign_function) states that, \frac{d}{dx}\vert x\vert = \sgn(x) and \frac{d}{dx}\sgn(x) = 2\delta(x). I'm wondering why the following is not true: \begin{align*} \vert x\vert &= x\sgn(x)\\ \Longrightarrow \frac{d}{dx}\vert x...

I'm trying to do a calculation which involves taking the derivative of a term that is millions of terms long. I'm using a supercomputer right now with Mathematica to do it but it's taking to long. Does Matlab have a faster algorithm for computing derivatives analytically?

You know that the problem of calculus of variations is finding a y(x) for which \int_a^b L(x,y,y') dx is stationary. I want to know is it possible to solve this problem when L is a function of also y'' ? It happens e.g. in the variational method in quantum mechanics where we say that choosing...

First off: I think I understand the chain rule and how it derives from \lim_{h \to 0} \frac{ f(x+h)-f(x)}{h} and how to apply the chain rule when taking the derivative of an implicit function. The textbook I am reading Applied Calculus (by B. Rockett) uses the following example on...

Hey guys, I have a couple of questions about this problem set I've been working on. I'm doubting some of my answers and I'd appreciate some help. Question: For 1a, using inverse trigonometric derivative identities should work, right? I got y' = 1/sinØ + 1/cosØ and multiplied by the common...

Integrate[f[qs] DiracDelta'[qs (1 - 1/x)], {qs, -\[Infinity], \[Infinity]}, Assumptions -> 0 < x < 1] Integrate[f[qs] DiracDelta'[qs - qs/x], {qs, -\[Infinity], \[Infinity]}, Assumptions -> 0 < x < 1] This is on Mathematica 8 for windows. The results differ by a sign. They are effective...

I know how to prove this via limits and I'm okay with that. What I want to understand is the interpretation of the theorem and specifically a visualisation of why what the theorem states must be the case. My guess is that this theorem is saying that change is symmetrical. But I don't know...

1. Marine biologists have determined that when a shark detects the presence of blood in the water, it will swim in the direction in which the concentration of the blood increases most rapidly. Suppose that in a certain case, the concentration of blood at a point P(x; y) on the surface of the...

Homework Statement Suppose p(x)\ =\ a_0\ +\ a_1x\ +\ a_2x^2\ +\ ...\ + a_nx^n. Now if |p(x)|\ <=\ |e^{x-1}\ -\ 1| for all x\ >=\ 0 then Prove |a_1\ +\ 2a_2\ +\ ...\ + na_n|\ <=\ 1. 2. Relevant Graph( |e^{x-1}\ -\ 1| ) The Attempt at a Solution From the graph we can conclude that p(x)...

Homework Statement I have to find the deriv of ##f(x)=xArctan(x^{3})## I just need an explanation of how the arctan works... So I understand the rest, but just the deriv of arctan itself is confusing for me. So the derivative of ##arctan(x)## just in general is ##\frac{1}{1+x^{2}}## But if...

i came to this topic and they said Duf(x) = ||gradient vector|| * ||U|| * cos 0 if ||U|| were not a unit vector it would give different rate of change of f in any direction what would happens if used ||U|| = 10 ?

Homework Statement Show that if σ(t) for (t in I) is a parametrization of a line, then σ''(t) is parallel to σ'(t). Homework Equations The Attempt at a Solution I thought that if σ(t) is a parametrization of a line then it could be expressed as σ(t) = vt + a, but then σ'(t) = v...

Homework Statement Hello, I'm having trouble understanding this, seemingly simple, concept. Any help or input is appreciated. Evaluate the following derivatives: $$\frac{d}{dt} u(t-1)u(t+1)$$ $$\frac{d}{dt} r(t-6)u(t-2)$$ $$\frac{d}{dt} sin(4t)u(t-3)$$Homework Equations The Attempt at a...

We can get a lot of good intuition for how first and second derivatives work by interpreting a sign restriction. Let ##I\subseteq \mathbb R## be an interval and ##f:I\to\mathbb R##. 1) If ##f## is differentiable, then ##f## is monotone iff ##f'\geq 0## everywhere. 2) If ##f## is twice...

Hi Guys, I am revising for an exam i have this week, the last module on my subject was calculus. I did not understand it entirely. I have posted a pic below of a typical problem i can expect to encounter, would anybody be able to point me in the right direction to study material that...

I've never really noticed whether this is true but if I know that: dt/ds = k/(1-s/r) for example, how does one find ds/dt? Is it the inverse, as you would expect, or is there some other method?

Need to find the Derivative using the chain rule y = x2sin4(x) + xcos-2(x) I am not sure where to start. answer in book is 2xsin4(x) + 4x2sin3(x)cos(x) + cos-2(x) +2xcos-3(x) xsin(x)

Problem: y = ln|sec(x) + tan(x)| Attempted Solution: See Attachment I was hoping someone could identify my error and possibly write it out for me. At first I thought my steps were correct and everything was in order. However, when I checked my answer on WolframAlpha it gave me a slightly...

I' at a a very very very basic level of calculus and usually have to watch a video or read something basic just to understand the basics. I'm fascinated by optimization equations, for example what is the largest area that can made with 500m of fencing. So at some point in solving this we end...

I am posting this question, in order to make something clear, since i am confused by derivation of the exponential function. I'll post the formula i used, correct me if you find something wrong thank you: {\frac{d}{Dx}}\ e^x => {\frac{e^{x + Dx} - e^x}{Dx}}\ => Here i factored out...

Need to find derivative f(s) = [(√s) -1]/[(√s) + 1] answer in book is f'(s) = 1/[√s(√s+1)^2]

Why is it that the covariant derivative of a covariant tensor does not seem to follow the product rule like contravariant tensors do when taking the covariant derivatives of those? Here is a visual of what I mean: This is the covariant derivative of a contravariant vector. As you can see, it...

Hi all, I came across the term placement, and presentment and the likes. I was wondering what they may mean in physics. I tried a search but the answers were not very clear. I was also wondering what action means in physics. Any answers will be appreciated. Thanks

find the indicated derivative dp/dq if p = 1/(√q+1) I apologize ahead of time if you can't read my work. my work [(1/(√(q+h+1))) - (1/(√(q+1))] \divh [((√(q+1)) - (√(q+h+1)))/((√(q+h+1))(√(q+1)))] \divh [(q+1-q-h-1)/(((√q+h+1)(√q+1))((√q+1)+(√q+h+1)))]\div h...

Hello. Can someone check if my partial derivatives are correct? I am not so confident with my answers.

Initially, the purpose of this tutorial will be to explore and evaluate various lower order derivatives of the Hurwitz Zeta function. In each case, the Hurwitz Zeta function will be differentiated with respect to its first parameter. A little later on - although this will take some time! - these...

Homework Statement If f and g are both analytic in a domain D and if f'(z)=g'(z) for every z\in{D}, show that f and g differ by a constant in D Homework Equations Cauchy-Riemann Equations Possibly Mean Value Theorem The Attempt at a Solution I'm pretty sure I'm making a...

I've come across a question which as really stumped me because I thought I knew how to do this but apparently not. The question is that we have a tangent vector to a level curve of a function of two variables f(x,y) at a point P is (2,1). Hence by my logic this means grad of f x unit vector...

Given V=xf(u) and u = \frac{y}{x} How do you show that: x^2 \frac{\partial^2V}{\partial x^2} + 2xy\frac{\partial^2V}{\partial x\partial y} + y^2 \frac{\partial^2V}{\partial y^2}= 0 My main problem is that I am not sure how to express V in terms of a total differential, because it is a...

Homework Statement Suppose z=ψ(2x-3y), Show that the second partial derivative of z with respect to x, is equal to the second partial derivative with respect to y multiplied by a scalar k. Homework Equations The Attempt at a Solution I thought this was too simple to be correct...

Help with this problem -- Proof with first and second derivatives Homework Statement I'm stuck on this problem and I'm not sure what I'm missing. The problem states: Assume that |f''(x)| \leq m for each x in the interval [0,a] , and assume that f takes on its largest value at an interior...

Suppose we have a function V(x,y)=x^2 + axy + y^2 how do we write \frac{dV}{dt} For instance if V(x,y)=x^2 + y^2, then \frac{dV}{dt} = 2x \frac{dx}{dt} + 2y \frac{dy}{dt} So, is the solution \frac{dV}{dt} = 2x \frac{dx}{dt} + ay\frac{dx}{dt} + ax\frac{dy}{dt} + 2y \frac{dy}{dt}

I have some complicated function f of the variables x,y: f(x,y) Now I can't really invert this expression for f for x and y, but I want the derivative of x and y wrt f. How can I do that? Am I allowed to say: ∂x/∂f = 1/(∂f/∂x) I have seen physicists "cheat" by using this relation, though I...

Can someone explain why the derivative with respect to a contravariant coordinate transforms as a covariant 4-vector and the derivative with respect to a covariant coordinate transforms as a contravariant 4-vector.

Hi guys, Question is: Find the slopes of the curves of intersection of surface z = f(x,y) with the planes perpendicular to the x-axis and y-axis respectively at the given point. z = 2x2y ...at (1,1). fx(x,y) = 4xy ∴ Slope = 4 fy(x,y) = 2x2 ∴ Slope = 2 Is this wrong? Answer...

What means: ? This guy, ##\vec{\nabla}_{\hat{\phi}} \hat{r}##, for example, means: \\ \hat{\phi}\cdot\vec{\nabla}\hat{r} = \begin{bmatrix} \phi _1 \\ \phi _2 \\ \end{bmatrix} \begin{bmatrix} \frac{\partial r_1}{\partial x} & \frac{\partial r_1}{\partial y} \\ \frac{\partial...

Given a vector field f, I can compute the rotational tendency in the direction n (∇×f·n), the translational tendency in the direction t (∇f·t) and the divergence (∇·f) too. So, given a scalar field f, why I can compute only the directional derivative (translational tendency (∇f·t)) in the wanted...

I have an easy question which I've been thinking about for a while.. Lets say I want to take the derivative of a function y = f(x) with respect to x, we would get. dy/dx = f'(x). In the couple of books I've skimmed through, they all say that dy/dx is not a ratio but the notation that...

Hello all, I have this function here: \[f(x,y)=\left\{\begin{matrix} z &(x,y)\neq (0,0) \\ 0 & (x,y)=(0,0) \end{matrix}\right.\] where \[z=\frac{x^{3}+xy^{2}}{2x^{2}+y^{2}}\] And I need to find it's first partial derivative by x and y at the point (0,0). I am not sure I know how to approach...

Function ##f(x)=x^4## has minimum at ##x=0##. ## f'(x)=4x^3## ##f'(0)=0## ##f''(x)=12x^2## ##f''(0)=0## ##f^{(3)}(x)=24x## ##f^{(3)}(0)=0## ##f^{(4)}(0)>0## So what is the rule? I must go to the higher derivatives if ##f'(0)=f''(0)=0##?

I'm trying to work out: (∇f)^2 (f is just some function, its not really important) While working in curved space with a metric: ds^2 = α dt^2 + dr^2 + 2c√(α+1) dtdr I'm not really sure how to calculate a derivative in curved space, any help would be appreciated thanks

Homework Statement Using the formal limit definition of the derivative, derive expressions for the Fourier Transforms with respect to x of the partial derivatives \frac{\partial u}{\partial t} and \frac {\partial u}{\partial x} . Homework Equations The Fourier Transform of a function...

Hey, I am actually a third year physics student, but here in New Zealand they tend to rush past the fundamentals, hence I can't seem to rearrange the following equations properly. Eq 1 and Eq 2 are: τ=-erB*dr/dt ek and τ=m*d(r^2dθ/dt)/dt ek Where ek is a unit vector. Is supposed to...

Good afternoon guys! I have some doubts about partial derivatives. The other day, my analytic geometry professor told us that slopes do not exist in three-dimensional space. If that's the case, then what does a partial derivative represent? Given that the derivative of a function with respect to...