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

learn more… | top users | synonyms (2)

3
votes
2answers
84 views

How to properly find supremum of a function $f(x,y,z)$ on a cube $[0,1]^3$?

Solving an applied problem I was faced with the need to find supremum of the following function $$f(x,y,z)=\frac{(x-xyz)(y-xyz)(z-xyz)}{(1-xyz)^3}$$ where $f\colon\ [0,1]^3\backslash\{(1,1,1)\} \to\...
2
votes
0answers
78 views

Functions $g(x)/h(x),h(x)/f(x)$ are constant [duplicate]

Suppose $f$, $g$, $h$ are functions from the set of positive real numbers into itself satisfying $f(x)g(y)=h(\sqrt{x^2+y^2})$ for all $x$, $y\in (0,\infty)$. Show that the functions $g(x)/h(x)$, $h(...
1
vote
1answer
19 views

Find the maximum value of the function

So I was just messing around with finding the maximum and minimum values of functions, and I came across this: $$ \text{Find the maximum value of} \,\, f(x)=\frac1{x^{2x^2}}.$$ Any ideas?
4
votes
3answers
47 views

Differentiability of function for $\Bbb{Q}$ and $\Bbb{R}\setminus \Bbb{Q}$

A function $f:\Bbb{R}\to\Bbb{R}$ is defined by $f(x)=x$, if $x$ is rational; $\sin(x)$ if $x$ is irrational. Show that $f$ is differentiable at $0$ and $f'(0)=1$. Here I'm thinking to apply ...
1
vote
2answers
23 views

$f(x, y) = \prod_{i = 1}^n (1 + xy_i)$, what is ${{{\partial f}\over{\partial x}}\over f}$, geometric series?

Let$$f(x, y) = \prod_{i = 1}^n (1 + xy_i).$$What is$${{{\partial f}\over{\partial x}}\over f}?$$What happens when we use the geometric series?
7
votes
2answers
137 views

Chain Rule and Vector valued functions?

Let $f: R^n \to R$ be given by $f(x) = \frac{||x||^4} {1 + ||x||^2}$ . Use the chain rule to show that $f$ is differentiable at each $x \in R^n$ and compute $Df(x)$. This vector valued stuff just ...
0
votes
0answers
26 views

Convergence of a function of two variables

The following question has been posed to me by a student in an analysis class. For which real numbers $\alpha \gt 0$ is the function $f : \Bbb R^2 \to \Bbb R$ given by $f(x, y) = (x^2 + y^2)^\alpha$ ...
0
votes
2answers
29 views

Prove that there exists some real number θ satisfying 0 < θ < 1 for which f '''(θ) = 0

Let f: D → R be a 3-times differentiable function defined over an open interval D, where 0 ∈ D and 1 ∈ D. Suppose that f(0) = f '(0) = 0 and f(1) = f '(1) = 0. Prove that there exists some real ...
-1
votes
1answer
26 views

Question about applying the Chain Rule with multiple variables

Let $z = u(x,y)$ and $y = y(x)$ and $u(x,y(x))$ = 0. What is the second derivative of the function $y(x)$? I tried to use chain rule but I keep making mistakes
0
votes
1answer
35 views

Why are scale factors not always unity?

A scale factor in curvilinear coordinates is defined as $$h_v \equiv \left|\frac{\partial\vec{r}}{\partial v}\right|$$ where $\vec{r}=(x,y,z)^T$ is a position vector. The partial differential can be ...
0
votes
2answers
54 views

What is y'' if $\sin y = y + 5x$?

I got $ 5\sin y / (\cos y - 1)^2$ as my answer, but the correct answer was given as $25\sin y / (\cos y - 1)^3$. My thought process: Derive the original equation to get $y'\cos y = y' +5$ $$y'(\cos ...
2
votes
3answers
72 views

$f:[0,1] \to \mathbb{R}$ is differentiable and $|f'(x)|\le|f(x)|$ $\forall$ $x \in [0,1]$,$f(0)=0$.Show that $f(x)=0$ $\forall$ $x \in [0,1]$

$f:[0,1] \to \mathbb{R}$ is differentiable and $|f'(x)|\le|f(x)|$ $\forall$ $x \in [0,1]$,$f(0)=0$.Show that $f(x)=0$ $\forall$ $x \in [0,1]$ I used the definition of derivative: $f'(x)=|\lim_{h \...
12
votes
4answers
705 views

Double obstructing wall problem, what is the optimal walk path and length?

Every day, you walk from point A to point B which are exactly $2$ miles apart straight line distance, however, each day, there is a $50$% chance of there being an obstructing wall perpendicular to the ...
2
votes
1answer
27 views

Study the absolute minima and maxima of $f(x,y)=(x^2-y^2)(x-2)$

Study the absolute minima and maxima of $$f(x,y)=(x^2-y^2)(x-2)$$ in the triangle $A$ of this vertices: $$O=(0,0) \qquad P=(2,2) \qquad Q=(2,-2)$$ I consider the set: $$A=\{ (x,y) \in \mathbb{R}^2 : ...
1
vote
2answers
33 views

Implicitly finding the derivative of $f^{-1}(x)$ given $f(x)$

Can we find the derivative of the inverse of a function implicitly by finding the derivative of the original function? For example lets say I have $f(x) = e^x$ and I want to find the derivative of ...
2
votes
1answer
37 views

Where i am going wrong in finding normal to curve?

The question is Find the perpendicular distance between the normal to the curve $$x=a\cos t+at\sin t, y=a\sin t-at\cos t$$ and the origin. Equation is given in parameterized form. My attempt ...
2
votes
2answers
31 views

Is there a solution to this differential equation?

I am trying to find a function $y(x)$ that is a solution to $$ \left(a_3 x^3+a_1 x\right) y''(x)-\left(3 a_3 x^2+2 a_1\right) y'(x)+3 a_3\, x \,y(x)=a_0 x^4+a_2 $$ I tried using mathematica but it ...
0
votes
2answers
9 views

Deduce that: $\frac{d}{dx}(u^{m}v^{n})=u^{m-1}v^{n-1}[mv\frac{du}{dx}+nu\frac{dv}{dx}]$

Deduce that: $$\frac{d}{dx}(u^{m}v^{n})=u^{m-1}v^{n-1}(mv\frac{du}{dx}+nu\frac{dv}{dx})$$ When I differentiate $\frac{d}{dx}(u^{m}v^{n})$ I get: $$\frac{d}{dx}(u^{m}v^{n})=u^{m-1}v^{n-1}(mv+nu)$$ Is ...
1
vote
1answer
61 views

How to find $\frac{\mathrm{d}y}{\mathrm{d}x}$ when both number in front and exponent have fractions?

I'm not sure how to solve this: $\frac{5}{9}x^\frac{2}{3}$. I applied the product rule and have $\frac{2}{3}\frac{5}{9}x^{-\frac{1}{3}}$. $\frac{30}{9}x^{-\frac{1}{3}}$, then $\frac{9}{30}x^{\frac{1}...
3
votes
1answer
37 views

Existence of Derivative for an Integral

Let $f$ be a Riemann integrable function defined on $[-2,2]$. Define a function $F \colon (-1,1) \to \mathbb{R}$ by $$F(h)=\int_0^1 h | f(x+h)-f(x)|\, dx.$$ Show that the derivative $F'(0)$ exists. ...
1
vote
1answer
36 views

Cannot make sense of a derivative

Short version of the question: In this presentation http://www.slideshare.net/ShangxuanZhang/xgboost (page 74-75)I cannot understand how the gradient of the L function is calculated. $$ L = y_i log ...
2
votes
1answer
41 views

Prove the exterior derivative of the following (n-1) form is zero

Let $\omega(x)=\frac{1}{{\parallel x \parallel}^n}\displaystyle\sum_{i=1}^{n}(-1)^{i-1}x_{i} dx_{1} \wedge \dots \wedge \widehat{dx_{i}} \wedge \dots \wedge dx_{n}$ be a differential $(n-1)$ form on $\...
1
vote
1answer
38 views

How do these partial derivative and derivative terms relate?

From the top line, this proof jumps to the integration and evaluation of the function. I'm not sure how the partial of $t$ and $dt$ play in the integration to give $(s,t)$ before evaluation. Any help ...
3
votes
1answer
39 views

Drones and Integrals Project

Hello everyone and thanks for taking the time to read this post. So in my college calculus class we had the opportunity to fly a drone and get it's flight data. I have a spreadsheet featuring two ...
1
vote
1answer
43 views

How to prove derivative of logarithm with base $b$?

I learned how to derive a logarithm with any base. This is the formula: $$\frac{d}{dx}\log_bx=\frac{1}{x\ln b}$$ How can it be proved?
0
votes
0answers
36 views

upper bound on derivatives of a function defined on an arc

Given a smooth arc on the complex plane by $z=\cos t + 0.5 i \sin t,\; t\in[\pi/10,\pi/5] $ , and a non-analytic function $f(z) = \text{Re } z $ defined on the arc. Obviously, $f(z) = g(t) :=...
1
vote
0answers
9 views

2D Fourier Differentiation in Matlab

I am working in Matlab to compute partial derivatives and a Laplacian using Fourier Transforms. I prefer to take spectral derivatives, since it corresponds to multiplication by a wave number, however, ...
2
votes
3answers
37 views

Minor flaw in understanding of the proof of the derivative of exponential functions

I understand the majority of the proof of the derivative formula for exponential functions of the form: (full proof at bottom of post) $\frac{d}{dx}a^x$ but I have a little trouble with the last ...
1
vote
3answers
34 views

Searching for a sequence of functions

Consider the following set of functions: $$ A=\left\{f\in C([0,1],\mathbb{R}): f(0)=0, \lim_{r\searrow 0}\frac{f(r)}{r}\text{ exists}\right\}. $$ Is there a sequence $(f_n)\in A^{\mathbb{N}}$ such ...
2
votes
2answers
61 views

Suppose $f$ is differentiable at $a$. Evaluate $\lim\limits_{h\to0} \frac{f(a+16h) - f(a+15h)}h$

Suppose $f$ is differentiable at $a$. Evaluate if possible $$\lim\limits_{h\to0} \frac{f(a+16h) - f(a+15h)}h$$ $$\lim\limits_{h\to0} \frac{f(a+15h)}h - \lim\limits_{h\to0} \frac{f(a+15h)}h$$ which to ...
1
vote
1answer
21 views

Differentiability of vector valued function?

The question is: Let $g$ and $h$ be real-valued functions on $R$ such that $g$ is differentiable at "a" and $h$ is differentiable at "b". Show that $f$: $R$2 → $R$ defined by $f$( x1, x2) = $g$(...
2
votes
2answers
65 views

Show that $\lim_{h\to 0}\frac{e^h-1}{h} = \ln e = 1$ using at least two numerical examples.

Show that $$\lim_{h\to 0}\frac{e^h-1}{h} = \ln e = 1$$ using at least two numerical examples. To solve this should I find numbers for $h$ that makes those equations equal to $1$? And how should I go ...
4
votes
3answers
91 views

Please prove the following: Given $ƒ(x) = e^x$, verify that $\lim_{h\to 0}\frac{e^{x+h} – e^x}{h} = e^x$.

Given $ƒ(x) = e^x$, verify that $$\lim_{h\to 0}\frac{e^{x+h} – e^x}{h} = e^x$$ and explain how this illustrates that $f'(x) = \ln e \cdot f(x) = f(x)$.
0
votes
0answers
20 views

Big 'O' Notation - Taylor Series

Q) Use the Taylor Series Expansion to show the first derivative f '(x) can be approximated by $$-(3f(x) -2f(x+h) - f(x-h) / h ) $$ What is the precision? Now I found after using the Taylor ...
0
votes
1answer
14 views

Taking the derivative of $\epsilon \cdot(\ln X + \ln \beta) - \ln(1 + X^{\frac{\alpha}{\alpha - 1}})$ with respect to $\ln X$

So I am taking the derivative of $$\epsilon \cdot(\ln X + \ln \beta) - \ln(1 + X^{\frac{\alpha}{\alpha - 1}})$$ with respect to $\ln X$, where $X$ is a variable, $\epsilon, \beta, \alpha$ are ...
0
votes
1answer
39 views

The region where the two variable function $xy/(x-y)$ is differentiable

I need to found the area where this function is differentiable $$ f(x,y) = \frac{xy}{x-y} $$ How do I need to proceed? For partial derivatives I got: $$ \frac{\partial f}{\partial x} = \frac{-y^2}{...
3
votes
1answer
57 views

Function that is second differential continuous

Let $f:[0,1]\rightarrow\mathbb{R}$ be a function whose second derivative $f''(x)$ is continuous on $[0,1]$. Suppose that f(0)=f(1)=0 and that $|f''(x)|<1$ for any $x\in [0,1]$. Then $$|f'(\frac{1}{...
2
votes
1answer
56 views

Sum of a polynomial with all its derivative [duplicate]

Let $$p(x)=x^n+a_1x^{n-1}+...+a_{n-1}x+a_n,$$ with $n$ is even and $p(x)>0$ for all $x\in\mathbb{R}$. Let $$q(x)=p(x)+p'(x)+..+p^{(n-1)}(x)+p^{(n)}(x).$$ Show that $q(x)>0$ for all $x\in\mathbb{...
1
vote
0answers
31 views

Let $\alpha$ be a real number. Find the value of $\alpha$ for which the given function is continuous and differentiable.

Let $\alpha$ be a real number. Consider the function $$g(x)=(\alpha+|x|)^2e^{(5-|x|)^2}, \ \ \ -\infty<x<\infty $$ $(i)$ Determine the values of $\alpha$ for which $g$ is continuous at all $x$. ...
0
votes
1answer
21 views

The upper bound of a sum of exponential function

Could someone help me to find the upper bound of the following function: $f(x) = \sqrt{\sum_{n=i}^{N} e^{-\alpha_{i}\cdot x}}$, where $x > 0$, the $i^{th}$ coefficient $\alpha_{i} > 0$. I got ...
0
votes
1answer
26 views

Find derivative of $4\cos(x - \frac{311\pi}{180})$

Find derivative of $4\cos(x - \frac{311\pi}{180})$ I think I should use the chain rule which states: $h'(x) = g'(f(x)) \cdot f'(x)$ $g'(f(x)) = - 4 \sin(4\cos(x - \frac{311\pi}{180}))$ $f'(x) = x$...
0
votes
2answers
21 views

Taking derivatives of the integration bounds

It has been a while since I took rudimentary calculus classes, so I might be slipping on the basics. I tripping on how to differentiate the lower and upper bounds of an integral. For example, lets ...
0
votes
0answers
13 views

Weak Derivative of logarithm

How can I show that $ln|x|$ on $(-1,1)$ has no weak derivative but in $B_1(0)\in R^2 $ it has? I know that every classical solution is also a weak solution in this case. But I don't how to show.
2
votes
1answer
34 views

Suppose $f$ is a mapping between a normed space and a Hilbert space with ONB $(e_n)_n$, what's the second derivative of $\langle f,e_n\rangle$?

Let $E$ be a normed space $(H,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a Hilbert space $f:E\to H$ be Fréchet differentiable $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ and $$f_n:=\langle f,...
1
vote
1answer
25 views

Suppose $f$ is a mapping between a normed space and a Hilbert space with ONB $(e_n)_{n\in\mathbb N}$, what's the derivative of $\langle f,e_n\rangle$?

Let $E$ be a normed space $(H,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a Hilbert space $f:E\to H$ be Fréchet differentiable $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ and $$f_n:=\langle f,...
0
votes
0answers
12 views

Gradient Vector of Homogeneous Functions

I've been given the definition $f:\mathbb{R}^n \to \mathbb{R}^m$ is homogeneous of degree k if $f(\lambda x)=\lambda^kf(x)$ $\forall x\in\mathbb{R}^n, \lambda>0$ and asked to show $<\nabla f(x),...
0
votes
1answer
14 views

Finding the modulus of complex functions

Let $\gamma$ be the path$$\gamma:\left[0,1\right]\rightarrow\mathbb{C}, t\rightarrow\exp\left(t+it\right)$$ I have found that $$\gamma'\left(t\right)=\left(1+i\right)\exp\left(t+it\right)$$ To find ...
0
votes
1answer
46 views

What does the symbol “$|_{\epsilon=0}$” mean with a derivative? [duplicate]

What does the notation "$|_{\epsilon=0}$" at the bottom of the derivative mean?
2
votes
1answer
22 views

Having trouble deriving the symbols used in a quadratric approximation problem.

I'm refreshing my calculus by studying MIT OCW's Single Variable Calculus course online. The problem is 2A-11, part of Unit 2 "Applications of Derivatives". It's a problem dealing with quadratic ...
-1
votes
1answer
72 views

How can I find $dy/dx$? [closed]

What does $dy/dx$ represent for these questions? $y = x^5$ $y = x+5$ $y = b$, $b$ is a constant Am I supposed to divide the $y$ by $x$? So, $\frac{y}{x^5}$ and $\frac{y-5}{x}$? If so, what do I do ...