Questions on the evaluation of derivatives or problems involving derivatives (for example, use of the mean value theorem).

learn more… | top users | synonyms (2)

0
votes
2answers
40 views

Finding the partial derivatives of $V (x, y) = U (x, y)e^{−ax−by}$

I think I did something wrong, so I was hoping someone might be able to show me the solution Two functions $V (x, y)$ and $U (x, y)$ are connected by the equation $$V (x, y) = U (x, y)e^{−ax−by}$$ ...
1
vote
1answer
64 views

Two methods of solving the differential equation $y' = .75 -.005y$

I am working on a differential equation problem and I am stumped since two different methods seem to give me two different answers Method 1 Given $\frac{dy}{dx} = .75 -.005y$ ...
0
votes
1answer
47 views

How would I use derivatives for suggesting an option to my user?

I was learning derivatives. I understood the theoretical concept behind it. When I was searching for the real-life example in machine learning I came across one of the answers in this question How do ...
0
votes
2answers
58 views

Find equation of tangent line to a curve $g(x)$ at $x=4$

So I am trying to find the equation of a tangent line to the curve: $$y = g(x)\text{ at }\,x = 4$$ given $g(4) = -6,\;$ and $\;g'(4) = 2$.
0
votes
2answers
208 views

Least Squares: Derivation of Normal Equations with Chain Rule

I'm new to Stackexchange so please bear with me. I'm struggling with the least squares formula. Now Wikipedia does show ways to derive the "normal equations". But I'd like to get the same result ...
0
votes
2answers
40 views

Distributional derivate of $f(t)$

I have the function $f(t)=e^{-|t|}$ And I want to distribution derivate it to $f''(t)$. I am aware of that the $f'(t)$ function will be: But how do I derivate to $f''(t)$ ?
1
vote
0answers
60 views

How can I calculate this matrix differentiation?

I am studying about the Matrix Differentiation. I don't know if this red box differential metric, which is how it is calculated.
3
votes
2answers
140 views

Does my proof of $|x+y| \le |x| + |y|$ make sense? How do I conclude a proof?

Thank you for reading it. I know I made a lot of mistakes. This is my first ever proof that I have attempted. Another note is that I only have been studying proofs for about a week. Any advice will be ...
2
votes
0answers
66 views

$n$th derivative of $f(x)$ using limit definition

After playing around with the limit definition of the derivative for higher order derivatives, I noticed the following odd relationship to determine it for an nth order derivative: Let $F^n=f(x+nh)$ ...
0
votes
0answers
21 views

Prove that the maximums of the family$f(x)=xe^{-ax},\ a>0$ are collinear.

If $f(x)=xe^{-ax}$, and a is an integer constant greater than zero, then $$\frac{df}{dx}=(1-ax)(e^{-ax}).$$ The maximum of $f(x)$ would then be at the $x$-value where $\frac{df}{dx}=0$. Since ...
2
votes
1answer
36 views

Prove that $f$ is differentiable in $(0,0)$ if and only if $\lim_{t\to0+} g(t)$ exists

Let $g:[0,\infty)\to\mathbb{R}$ be a mapping and $f(x,y)=xg(\sqrt{x^2+y^2})$ for all $(x,y)\in\mathbb{R^2}$. Prove that $f$ is differentiable in $(0,0)$ $\iff$ $\lim_{t\to0+} g(t)$ exists. My ...
2
votes
9answers
275 views

Why does $(a+b)^2= a^2+b^2 + 2ab$? Why is the $2ab$ there?

When I was doing research on finding the derivative I came across something strange. If $f(x) = x^2$ you find the derivative by going $$\frac{f(x+h)^2-f(x)^2}{h} =\frac{x^2+2xh+h^2-x^2}{h}.$$ Why ...
0
votes
0answers
26 views

n-th derivative of product

I am looking for a closed-form for the $n^{\text{th}}$ derivative of $$\beta_1 g_1 g_3 + \beta_2 g_2 g_3$$ if $$g_k^\prime = \alpha_k g_{k+1}$$ Here's what I have tried so far: \begin{align*} ...
1
vote
2answers
22 views

Special vs. General Case in Basic Algebraic Notation

From Rogawski, Jon (2011-04-01). Single Variable Calculus (Page 343). W. H. Freeman. : To compute the derivative of a function of the form eg(x), write eg(x) as a composite eg(x) = f(g(x) ), where ...
8
votes
2answers
402 views

Is there an easier way to find the “natural” integration constant?

Suppose we take consequtive derivatives of a function at a point and then interpolate them with Newton series (Newton interpolation formula) so to obtain a smooth curve. ...
0
votes
1answer
17 views

Showing the following differential equation is exact

I'm asked to show that the attached differential equation is exact: link. I know I have to show that Nx=My. In this particular equation, M = -x/siny - 2 and N = ((x^2+1)cosy)/(1-cos2y), and all I ...
0
votes
0answers
40 views

Proving concavity for complicated function

I have a rather complicated function, $f$, that I am trying to demonstrate is log-log-concave, i.e., $$\frac{d^2\log f}{(d\log x)^2}\leq 0.$$ The reason I think it is concave is purely heuristic. ...
1
vote
2answers
68 views

How to check for convexity of function that is not everywhere differentiable?

I have a question. I have just been introduced to the subject of convex sets and convex functions. I read this in wikipedia that a practical test for convexity is - to check whether the 2nd ...
1
vote
2answers
54 views

Prove that $\lim_{x\to\infty}f''(x)=0$ if $\lim_{x\to\infty}f(x)=T$ and $\lim_{x\to\infty}f'''(x)=0$.

Suppose that $f$ is a real function such that $f(x) \to T$, where $T$ is a finite limit, and that $f''' \to 0$ as $x \to \infty$. Prove that $f''(x) \to 0$ as $x \to \infty$.
2
votes
2answers
49 views

Find volume of cask

I was given the following question: A wine cask has a radius at the top of $30 cm$ and a radius at the middle of $40 cm$. The height of the cask is $1m$. What is the volume of the cask in litres, ...
3
votes
2answers
126 views

Find arc length of curve on the given interval

I was asked to find the arc length of the curve of the following curve: $24xy = x^4 + 48$ from $x = 2$ to $x = 4$ This has turned out to be a very difficult problem, I get stuck using the arc length ...
1
vote
2answers
87 views

Assumptions that can be made for $f(x) + xf '(x)\leq 0$

I am wondering if we can make any assumptions about a function $f$ i.f.f. it satisfies $$f(x) + xf '(x)\leq 0 \qquad\forall \;x>0\;?$$
4
votes
1answer
45 views

Deriving Laplace Transform of Laguerre polynomial

I'm given this definition for the Laguerre polynomials: $$L_n(t)=\frac{e^t}{n!}\frac{d^n}{dt^n}\left[t^ne^{-t}\right],~\text{for }n=0,1,2...$$ and I have to show that the Laplace transform is ...
1
vote
0answers
81 views

Verification: Hessian of the following composition.

I was hoping that someone could verify the steps of computing a Hessian matrix. I have the following function, $F:\mathbb{R}^n\to\mathbb{R}$, $$F({\bf x}) = \sum_{i=1}^mf(g(A_i^T{\bf x}))$$ where ...
4
votes
2answers
46 views

Implementing trig functions for dual numbers

I'm curious, how do common trig functions get implemented for dual numbers? One way would be to use the power series definition, but that seems inefficient
0
votes
1answer
70 views

Using the implicit function theorem to solve for two of four variables in the system of two equations

Show that there are positive numbers $p$ and $q$ and unique functions $u$ and $v$ from the interval $(-1-p, -1+p)$ into the interval $(1-q, 1+q)$ satisfying $$xe^{u(x)} +u(x)e^{v(x)}=0=xe^{v(x)} ...
1
vote
2answers
45 views

Derivative of certain piecewise function

I've got function $f(x) = \begin{cases} \frac{ln(1+x)}{x} &\text{for $x\not=0$} \\ 1 &\text{for $x=0$} \end{cases}$ And I need to find the derivative. (also one sided) I've found that ...
1
vote
4answers
66 views

Finding $p'(0)$ for the polynomial of least degree which has a local maximum at $x=1$ and a local minimum at $x=3$

The question goes as follows: Let $p(x)$ be a real polynomial of least degree which has a local maximum at $x=1$ and a local minimum at $x=3$. If $p(1)=6$ and $p(3)=2$, then $p'(0)$ is... What I ...
0
votes
2answers
71 views

Higher-order derivative test

Let $f:I\rightarrow \Bbb{R}$ $2007$ times differentiable at $x_0 \in I$. Also: $f'(x_0) = f''(x_0) = ... = f^{(2006)} = 0$ but $f^{(2007)} > 0$. Prove there's $\delta> 0$ such that $f$ is ...
0
votes
2answers
320 views

Simple differentiation / economics marginal cost question

This seems like a very simple question, so I'm sure I'm doing something stupid here, but I'm not quite getting my head around the following question: I have a total cost function: $C = 5x^2 +15x + ...
0
votes
0answers
29 views

Matrix Algebra - Linear dependency

We have a given equation $ \frac{\mathrm{d}R(t) }{\mathrm{d} t}=R(t) \{(1-t)U_0+t U_1\}\tag 1$, all variables except scalar variable 't' has dimension $3 \times 3$. Given data $R(t)$ is ...
1
vote
1answer
61 views

Understanding the notation of a book when derivating

I'm trying to understand the notation that the book uses. The book says $(1)$ $y=a\cdot \sin x$ and then the derivate of $(1)$ is $(2)$ $\frac{d^2y}{dx^2}=-a \cdot \sin x$ I don't get what to do ...
2
votes
0answers
92 views

Definite Integral involving matrices

We have a definite integral of the form given below $ f(t) = \int_0^1 e^{\alpha X(t)} \frac{dX(t)}{dt} e^{(1-\alpha) X(t)}\,d\alpha \tag 1$ Given Data in the question $X(t)$ is a ...
3
votes
3answers
159 views

Explanation of line element formula $dl^2 = dx^2 + dy^2$

I found this in a physics textbook without justification: $$dl^2 = dx^2 +dy^2,$$ where I presume that $l = \sqrt{x^2+y^2}$. Why is this so? By my calculations I obtain $$ dl = \dfrac{\partial ...
2
votes
4answers
146 views

Confusion about implicit differentiation.

I want to implicitly differentiate $Ax^2 + By^2 + Cxy + Dx + Ey + F = 0$. This is not an exceedingly difficult task, and when I solved it I got $$ y' = -\frac{2Ax + Cy + D}{2By + Cx + E} $$ But my ...
3
votes
4answers
203 views

Derivability of a piecewise function

Let's say I have a continuous piecewise function of a single variable, so that $y = f(x)$ if $x < c$ and $y = g(x)$ if $x>=c$. Is it right to say that the derivative of the function at $x=c$ ...
2
votes
3answers
97 views

Finding the second derivative of an infinite series

I'm asked to find the 2nd derivative of $$f(x)=-2x+\frac{2x^3}{3!}-\frac{2x^5}{5!}+\cdots+(-1)^{n+1}\frac{2x^{n+1}}{(2n+1)!}+\cdots=\sum \limits_{n=0}^{\infty} \frac{(-1)^{n+1}2x^{2n+1} }{(2n+1)!}$$ ...
1
vote
1answer
42 views

Difference quotients are increasing for $f$ convex

Problem 11A.9(c) in Spivak's Calculus (4th edition) asks the following (I'm paraphrasing): Suppose $f$ is convex. Show that $f'(a)$ exists iff $f_+'(x)$ is continuous at $a$. ($f_+'(x)$ is the ...
9
votes
2answers
221 views

$f'$ exists, but $\lim \frac{f(x)-f(y)}{x-y}$ does not exist

Suppose $f$ is differentiable at $a$, i.e. $\lim_{x\to a}\frac{f(x)-f(a)} {x-a}$ exists. I wondered whether it was necessarily true that $$\lim_{\substack{x,y\to a\\x\neq y}}\frac{f(x)-f(y)}{x-y} ...
1
vote
1answer
22 views

Question concerning an example of higher derivatives and binomes

I got this exercise form OCW 18.03SC - problem 1G-5b: What is the solution of the following derivative?: $$\dfrac{d^{p+q}}{dx^{p+q}}x^p(1+x)^q$$ I used Leibniz' formula and the only non-zero term ...
0
votes
1answer
47 views

ODE with Laplace transform: the jump of $\dot y$

I solved this eq. using the Laplace Transform: $\ddot y+4\dot y+13 y=\delta(t-2\pi)-\delta(t-7\pi)$ The sol. is: $y(t)=\frac{1}{3} e^{2 t} (-e^{14 \pi} \theta(t-7\pi) sin(3 t)+e^{4 \pi} \theta(t-2 ...
1
vote
2answers
80 views

How to prove that a diffrensiation of a formula equals to another formula.

QUESTION 1) if $y =\dfrac{ \sin x-x\cos x}{x\sin x+\cos x}$ show that $\dfrac{dy}{dx}= \dfrac{x^2}{(x\sin x+\cos x)^2}$ QUESTION 2) if $y = \dfrac{\tan x+1}{\tan x-1}$ show that $\dfrac{dy}{dx}= ...
0
votes
0answers
39 views

Use of the anti-derivative

Given a velocity function $dx/dt$, I am asked to find when a certain particles changes direction. This would then be when $dx/dt=0$. Let's say that $dx/dt$ has roots at $t= -1$ and $ t = 3$. I am ...
0
votes
4answers
72 views

Derivative of $\frac{1}{(x+1)^{k-1}}$

How is it that the derivative of $\frac{1}{(x+1)^{k-1}}$ is $-\frac{k-1}{x^k}$ where $k$ is a parameter.
0
votes
2answers
30 views

Arithmetic simplification

I am asked to find $\frac{d^2y}{dx^2}$, and prove that $\frac{d^2x^2+y^2=a^2}{dx^2}$=$-\frac{a^2}{y^3}$, This is how I have proceeded: $2 y \frac{dy}{dx}+2 x=2 a \frac{da}{dx}$ => ...
0
votes
1answer
30 views

Show that $f$ is a contraction if and only if there exists $r \in (0, 1)$ such that $|f'(x)| \leq r$, for all $x \in [a, b]$.

Let $f : [a, b] \rightarrow [a, b]$ be differentiable. Show that $f$ is a contraction if and only if there exists $r \in (0, 1)$ such that $|f'(x)| \leq r$, for all $x \in [a, b]$. I managed ...
0
votes
0answers
38 views

Differentiating integral function

I need to take the partial derivative of a function for the purpose of finding a zero with Newton Raphson. The function to be zero-ed is $R=\frac{\int_{\phi \ge 0} Y(\phi)d'(\phi)d\phi }{\int_{\phi ...
0
votes
1answer
174 views

derivative B-spline with own knot set

Define the spline function of degree $q$ on the interval $[\xi_0,\xi_K]$ $$f(t)=\sum_{j=1}^{K+q}b_j B_j(t)$$ where $B_j$ are degree $q$ B-spline basis functions determined by the knots ...
1
vote
1answer
30 views

Find the largest $n\in \Bbb{N}$ answering the following terms

Let $$f(x) = -\frac{1}{12}x^4 + o(x^5)$$ Also, Let $$g(x) = \begin{cases} \frac{f(x)}{x^n} &\mbox{if } x\ne 0 \\ C & \mbox{if } x=0 \end{cases}$$ I need to find the largest $n\in\Bbb{N}$ ...
0
votes
0answers
11 views

Given a matrix $X$ what is the gradient of $log|X|$

Given a matrix $X$ Is it possible to obtain the gradient of $log|X|$? That is $\frac{\partial log|X|}{\partial X}$