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

learn more… | top users | synonyms (2)

4
votes
2answers
86 views

How do I prove that the $f(x)$ is positive for all real $x$?

$$ \frac {f(x+y) - f(x)}{2}= \frac{f(y)-a}{2} +xy $$ for all real $x$ and $y$. If $f(x)$ is differentiable and $f'(0)$ exists for all real permisible values of $a$ and is equal to $\sqrt{5a-1-a^2}$. ...
0
votes
1answer
48 views

How to differentiate complex functions like this one?

If $$ y=\frac{1}{\sqrt{a^2-b^2-c^2}}\cos^{-1}{\left(\frac{at-a^2+b^2+ c^2}{t\sqrt{b^2+c^2}}\right)} $$ and $t=a+b \cos x+c \sin x$ prove that $\displaystyle\frac{dy}{dx}=\frac{1}{t}$.
-2
votes
1answer
77 views

$f(x)=2x^4+x^4\sin(\dfrac 1x) , \forall x \ne 0 ; f(0):=0$ ; it's derivative has both positive and negative values in every neighbourhood of $0$?

Let $f:\mathbb R \to \mathbb R$ a function defined as $f(x)=2x^4+x^4\sin\left(\dfrac 1x\right) , \forall x \ne 0 ; f(0):=0$ ; then how to show that it's derivative has both positive and negative ...
1
vote
1answer
198 views

Measurability of Dini Derivatives

Let $f:(0,1)\to\mathbb R$ be measurable. Then, the (right upper) Dini derivative $$ D^+ f(x) = \limsup_{h\to 0^+} \frac{f(x+h) - f(x)}{h} $$ is also measurable (a well known result of Banach). Can ...
1
vote
2answers
312 views

Increasing and decreasing piecewise function on an interval

I'm working on a problem that involves finding the intervals where a function $f$ is increasing and decreasing. Given the function$$ f(x) = \cases{ x+7 & \text{if } x\lt -3\cr |x+1| & \text{...
0
votes
1answer
44 views

Question about the derivative of $x \mapsto \langle x , x \rangle $ (Scalar product)

$f: x \mapsto \langle x , x \rangle $ In my book it says that $f'(x)=\langle 2x, \rangle$. I'm aware that $f: X \to \mathbb R$, what is not clear is this result. Can anyone explain how it comes to ...
0
votes
3answers
95 views

Finding $\frac{ \mathrm{d}y}{ \mathrm{d}x}$ when given that $\sqrt{1-x^2} + \sqrt{1-y^2} ={a}(x-y)$

Finding $\frac{ \mathrm{d}y}{ \mathrm{d}x}$ when given that $\sqrt{1-x^2} + \sqrt{1-y^2} = a(x-y)$. $a$ is a constant. I have the final answer, which is $$\frac{ \mathrm{d}y}{ \mathrm{d}x} = \sqrt{\...
1
vote
3answers
73 views

Checking the differentiability of piece-wise defined functions

I'm having trouble checking if a function is differentiable if it has a different definition for $x=0$. Here's an example: $$f(x)=\begin{cases} 0 &x=0 \\ x\sin(\frac{1}{x}) & x\ne 0 \end{...
0
votes
1answer
54 views

How do you go about solving partial differential equations for finding critical points in general optimization problems?

I was reading about partial second derivative test for optimization problems and I came across the example here. I saw the equations have yielded four critical points, but I wasn't able to find those ...
1
vote
1answer
86 views

Prove that continuity in x of the Gateaux derivative, $f'(x;y)$, implies Frechet differentiability

Prove that continuity in x of the Gateaux derivative implies Frechet differentiability Let $x$ be te point, $y$ the direction and $f'(x;y)=y·a(x)$. First, I considere the function $g(\varepsilon)=f(...
0
votes
1answer
19 views

If the tangents at

If the tangents at $P(1,1)$ on the curve $y^2 =x(2-x)^2$ meets the curve again at $Q$ then points of $Q$ is of the form $(3a/b,\, a/2b)$ so I have to find $a$ and $b$.
0
votes
1answer
25 views

Basic differentiation question on derivative of conical volume

So I was reading Polya's book and in it, there was a problem involving finding the rate of change of depth of water in a cone. At some point, we come to the conclusion that V = $\pi a^2 y^3/(3b^2)$ ...
2
votes
2answers
76 views

Derivative of a function with respect to x containing integral over y

does anyone know how to take a derivative of a function with respect to a variable if that function contains an integral over another variable? For example, what would be the derivative of the ...
1
vote
0answers
37 views

Completing proof of derivative being continuous

Suppose that $f$ is continuous at $a$, and that $f'(x)$ exists for all $x$ in some interval containing $a$, except perhaps for $x=a$. Suppose, moreover, that $\lim \limits_{x \to a} f'(x)$ exists. ...
1
vote
0answers
53 views

derivative of a tensor A with respect to transpose(A)*A?

What is the derivative of $\partial A/\partial ({A^T}A)$ ? Where $A$ is a 3x3 tensor. (in index notation, I want to find explicit components of ${D_{ijpq}} = \partial {A_{ij}}/\partial ({A_{kp}}{A_{...
3
votes
3answers
178 views

The range of the function $f:\mathbb{R}\to \mathbb{R}$ given by $f(x)=\frac{3+2 \sin x}{\sqrt{1+ \cos x}+\sqrt{1- \cos x}}$

The range of the function $f:\mathbb{R}\to \mathbb{R}$ given by $f(x)=\frac{3+2 \sin x}{\sqrt{1+ \cos x}+\sqrt{1- \cos x}}$ contains $N$ integers. Find the value of $10N$. I tried to find the minimum ...
4
votes
2answers
67 views

Find the maximum value of $72\int\limits_{0}^{y}\sqrt{x^4+(y-y^2)^2}dx$

Find the maximum value of $72\int\limits_{0}^{y}\sqrt{x^4+(y-y^2)^2}dx $ for $y\in[0,1].$ I tried to differentiate the given function by using DUIS leibnitz rule but the calculations are messy and I ...
3
votes
1answer
84 views

Is there a concept already for $\frac{f'(x)}{f(x)/x}$?

Given a differentiable real function $y=f(x)$, is there a math concept/terminology already defined for $$\frac{f'(x)}{f(x)/x}?$$ This quantity is inspired from price elasticity of demand. Thanks.
0
votes
0answers
92 views

Single variable optimization

A retail outlet for calculators sells 700 calculators per year. It costs \$2 dollars to store one calculator for a year; to reorder, there is a fixed cost of \$7 dollars plus \$2.25 for each ...
1
vote
3answers
75 views

Proof involving Rolle's theorem and the MVT

I'm stuck on a problem that asks me to show that the equation, $$6x^4-7x+1=0$$ does not have more than two distinct roots. I've tried to set up a proof by contradiction using Rolle's theorem and the ...
2
votes
1answer
37 views

$ \text{If } f,g \in D(U) \implies \alpha f + \beta g \in D(U)(\alpha f + \beta g)'(x)=\alpha f'(x) + \beta g'(x)$

Prove: $f,g$ are differentiable functions on open set $U \implies \alpha f + \beta g$ is differentiable on $U$ as well. Furthermore, $(\alpha f + \beta g)'(x)=\alpha f'(x) + \beta g'(x)$. Proof: We ...
3
votes
2answers
37 views

Maximizing Theta in a Summation Formula

I need to take the first derivative of $$\sum Y_i (\log(\Theta )) +(n-\sum Y_i)(\log(1-\Theta )) $$ with respect to theta, and then solve for theta. I believe this is my derivative... $$\frac d{d\...
2
votes
5answers
277 views

Prove a function is not differentiable using continuity

Given the function $f(x) = |8x^3 − 1|$ in the set $A = [0, 1].$ Prove that the function is not differentiable at $x = \frac12.$ The answer in my book is as follows: $$\lim_{x \to \frac12-} \dfrac{f(...
0
votes
0answers
32 views

Recommend a Maths Textbook for Calculus [duplicate]

I am not a beginner to calculus. Till know I have learned about: 1. Functions - Domain,range,odd/even,periodic,compositemapping, etc. 2. Graphical transformations. 3. Evalution of limits, checking ...
0
votes
0answers
29 views

cross product of material derivative

I am looking to evaluate $\vec{n} \times \dfrac{D\vec{u}}{Dt}$ where $\dfrac{D}{Dt}$ is the material derivative. Can I bring the cross product into the derivative and rewrite the expression as $\...
0
votes
1answer
35 views

Finding the derivative of a multivariable function

Suppose $f: \mathbb{R}^n \to \mathbb{R}$ is a differentiable function. Then we can write the derivative of $f$ as a $1 \times n$ row matrix of partial derivatives of $f$ ,i,e, $$Df=\begin{bmatrix}\...
3
votes
1answer
52 views

How to derivative the linear equation of matrix

I have the equation as $$F(w,x)=\sum_{i=1}^{N}\int_{x \in \Omega} \left ( Y(x)-w^TA(x)\right)^2u_i(x)dx$$ In which, $w$ is column vector that independent on $x$, denotes $w=[w_1,w_2...,w_M]^T$ $A$ ...
3
votes
5answers
78 views

What rule can I use to compute $\frac{d^{107}}{dx^{107}} \sin x$?

Did I miss something in my calculus class? I don't remember anything concerning this type of problem: Compute $$\frac{d^{107}}{dx^{107}} \sin x.$$ So what is the rule here?
3
votes
1answer
121 views

Differentiating composition of functions proof help

Theorem: Let $X, Y, Z$ be normed spaces and $U\subset X$, $V \subset Y$ open sets. If the function $f:U \to V$ is differentiable in $x \in U$ and function $g: V \to Z$ differentiable in $f(x)\in V$, ...
1
vote
2answers
56 views

Maximum value of the product of probabilities

I came across a confusing probability problem. It reads as follows: Let $S$ be a sample space and two mutually exclusive events $A$ and $B$ be such that $A \cup B = S$. If $P(\cdot)$ denotes the ...
1
vote
1answer
50 views

Good reference on higher dimensional derivatives?

I've spent several months now periodically scouring the internet for a comprehensive overview of an introduction to higher dimensional derivatives. I've already read baby Rudin's section on the topic,...
0
votes
1answer
77 views

Number of real roots of polynomial derivative

Let $W(x)$ be a polynomial with n real roots and $P(x) = \alpha W(x) + W'(x)$. Prove that for any $\alpha \in \mathbb{R}$: $P(x)$ have at least $n-1$ real roots. I know that the degree of the ...
4
votes
2answers
105 views

“Mean value like” problem.

Let $f:\mathbb{R} \longrightarrow \mathbb{R}$ be differentiable, take $a<a'<b<b'$. Prove that there exists $c<c'$ such that $$\frac{f(b)-f(a)}{b-a}=f'(c) \quad and \quad \frac{f(b')-f(a')}{...
4
votes
2answers
100 views

From Gravity Equation-of-Motion to General Solution in Polar Coordinates

I'm having trouble getting the general solution of this differential equation. The gravitational equation of motion is, for constants $M$ and $G$ and position vector $\vec{r}$, $$\frac{d^2}{d t^2}\...
0
votes
2answers
56 views

Finding conditional extrema with trig functions

Find the conditional extrema of $$f(x,y)=\cos^2x+\cos^2y,\quad g(x,y)=x-y+\frac{\pi}{4}=0.$$ I have a problem with finding a solution to this problem. Using Lagrange multipliers i come up with a ...
0
votes
1answer
67 views

Engineering Mathematics problem with proving an equation

This is problem 20, further problems in Engineering Mathematics book by K.A.Stroud. It states: Show that the equation \begin{equation} 4\frac{d^2x}{dt^2} + 4\mu\frac{dx}{dt} + \mu^2x = 0 \end{...
1
vote
3answers
41 views

How to verify my solution to an separable differential equation?

I have this question: Find the general solution to the separable differential equation $$ \frac{dy}{dx} = y(1-y). $$ My attempt is : $$ \frac{dy}{y(1-y)} = dx $$ $$ \frac{1}{y(1-y)} = \frac{A}{y}+...
2
votes
1answer
21 views

Problem with finding Maximum value

My problem states: Show that y: \begin{equation} y = e^{-t}sin(2t) \end{equation} is a maximum when \begin{equation} t = \frac{1}{2}\tan^{-1}(2) \end{equation} and determine this maximum value. So ...
0
votes
1answer
65 views

Question on continuity and differentiability of min() and max() functions.

Question: $f(x)=x^2-2|x|$. Test the continuity of $g(x)$ in the interval $[-2,3]$ if $g(x)$ is defined as: attempt: $f(x)$ is defined as: But i am finding it difficult to understand $g(x)$. ...
0
votes
1answer
36 views

How to compute high order differential?

Let $f: E \to \mathbb{R}$ sending $x \mapsto \|x\|$ and make some simple hypothesis $E$ is a Hilbert Space Let's say that the norm $\|\cdot\|$ is derived from a scalar product So we can easily ...
2
votes
0answers
32 views

derivative of a linear operator

I am a little confused by the proof in the picture. Doesn't the calculation show $dB(u)h = B(h_1,u_2)+ B(u_1, h_2)$?
0
votes
0answers
47 views

Definition third- and fourth-order partial derivatives

I have a real-valued function $f$ of a vector-valued variable $x$, i.e., $f:R^d\rightarrow R$. I need to define third- and fourth-order derivatives of $f$ in matrix notation fashion (regrettably, I ...
1
vote
1answer
48 views

Finding the Zeros

A word problem gives this cost equation and asks to find the x where the average cost is minimized, to do so, I need to solve for average cost and derive it and then set it to zero and solve; $c(x) = ...
0
votes
1answer
44 views

Question involving the PDF of a function of a random variable.

I'm trying to understand the setup for problem 3.1, from M.G. Bulmer's Principles of Statistics (Dover, 1967). Suppose that $X$ is a continuous random variable, and that $Y$ is a linear function ...
3
votes
1answer
78 views

How to find unkown height of triangle without hyptenuse

I been trying to solve this question and have tried to solve it for many days, but do not know how, any help would be much oblidged. A cable company owns the roads marked with the dotted lines in ...
0
votes
2answers
89 views

Help me finding nth derivate

I am beginning in Successive Differentiation. This is very simple question in differentiation, but don't where i am confused. Find a closed formula for the $n^{th}$ derivative of: $$\frac {x^2}{(x+...
2
votes
1answer
230 views

find the total differential of this equation $ xyz + \sqrt{ x^2 + y^2 + z^2} = \sqrt 2 $

How to calculate the total differential of $ z= z(x,y)$, which is $ xyz + \sqrt{ x^2 + y^2 + z^2} = \sqrt 2 $ at point (1, 0, -1)? The evaluation of mine seems wrong, $ dz= \frac{\partial z}{\...
0
votes
0answers
59 views

Derivative of a summation series

Let $f_{n}(x)=x+(1-x)x^2+(1-x)(1-x^2)x^3+\cdots+(1-x)(1-x^2)\cdots(1-x^{n-1})x^n;\quad n\geq4$ then $f'(x)=\text{ ?}$ $$(A)\qquad (1-f_{n}(x)) \left(\sum\limits_{r=1}^n \frac{rx^{r-1}}{1-x^r}\right)$$...
0
votes
2answers
66 views

Finding the Zeroes of a Second Derivative to Determine Points of Inflection

I have been told to graph the function $y=\cos \left(x^2\right)$ for $-2π ≤ x ≤ 2π$. I have determined key features of the graph but need help when it comes to determining the points of inflection for ...
2
votes
1answer
31 views

Minimizing long equation with hyperbolic functions

In physics book that I am reading it is said that minimizing the expression $$\phi = - N T k \log (2 \cosh(H \beta)) - \frac{J N}{2} z \tanh^2(H \beta) + H N \tanh(H \beta) $$ with respect to $H$ ...