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

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4
votes
4answers
113 views

$n^{th}$ derivative of $y=x^2\cos x$

I am stuck with Leibniz formula $$D^{n}y = \sum_{k=0}^{n} \binom{n}{k} \, x^{(2k)}\cos^{(n-k)}x$$ Could someone show how to do it?
0
votes
0answers
25 views

Derivatives of KL Divergence

We know that KL divergence $D(P||Q) = \sum_{i} p_{i}\log(\frac{p_{i}}{q_{i}})$, where P and Q are vectors. So, I think the derivatives of $D$ with respect to $Q$ is $\frac{\partial D}{\partial ...
0
votes
1answer
34 views

Completeness of space of $k$-times differentiable functions from $\Bbb R^n$ to $\Bbb R$

I'm having trouble solving exercise 7.5 from Otto Forsters Analysis 2 book: Let $\displaystyle U \subset \mathbb{R}^n$ be an open subset and $C^k_b(U)$ the set of all $k$-times continuously ...
0
votes
1answer
26 views

Need help with this proof, theory of finite change.

Theory: If $f : [a,b] \to X $ is differentiable on (a,b) and continuous on [a,b] in $X$, a normed vector space upon $ \langle , \rangle$ then: $$|f(b)-f(a)| \leq \sup_{a <c<b}{\|f'(c)\|}(b-a)$$ ...
4
votes
2answers
97 views

Finding the limit $\lim_{x\rightarrow 0^{+}}\frac{\int_{1}^{+\infty}\frac{e^{-xy}\quad-1}{y^3}dy}{\ln(1+x)}.$

Finding the following limit:$$\lim_{x\rightarrow 0^{+}}\frac{\int_{1}^{+\infty}\frac{e^{-xy}\quad-1}{y^3}dy}{\ln(1+x)}.$$ To my way of thinking,L'Hopital's rule is useful to this question.Then ...
3
votes
2answers
66 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
40 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
64 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 ...
0
votes
0answers
10 views

Qualitative comparison of parameterized sigmoid function (logit)

Having the logistic function to describe a 2 Alternative Forced Choice experiment: $f(x) = \frac{1}{1+e^{(a+bx)}}$ which from the results, I'm getting different sigmoids with different steepness. So ...
1
vote
0answers
26 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
33 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| & ...
0
votes
1answer
36 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
91 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} = ...
1
vote
3answers
56 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 ...
0
votes
1answer
30 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
40 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 ...
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
24 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 = $\pi a^2 y^3/(3b^2)$ ...
1
vote
2answers
20 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
31 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
20 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 ...
4
votes
2answers
61 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
77 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
13 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 ...
0
votes
1answer
43 views

Differentiability of $\frac{x^2y^2}{x^2+y^4}$ at $(0,0)$ [closed]

Given function, $f$ defined: $f(x,y)=\frac{x^2y^2}{x^2+y^4}$ if $(x,y)\ne (0,0)$ and $f(0,0)=0$ Prove that $f(x,y)$ is not differentiable at $(0,0).$
2
votes
4answers
67 views

Derivative of $e^{\sin(\ln[2 \arctan(2x)])}$ [closed]

Let $$F(x) = e^{\sin(\ln[2 \arctan(2x)])}$$ and take the first derivative of $F(x)$. Could someone walk me through each step here? It seems to be a pretty straightforward chain rule problem but it ...
0
votes
3answers
52 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
32 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
31 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 ...
2
votes
5answers
222 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-} ...
0
votes
0answers
26 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
8 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
21 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, ...
3
votes
1answer
41 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
66 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
105 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 ...
1
vote
2answers
40 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
43 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 ...
0
votes
1answer
58 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
102 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 ...
4
votes
2answers
63 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 ...
0
votes
2answers
26 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
57 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 ...
-1
votes
2answers
54 views

How do I prove the following equality of second and first order derivatives? [closed]

If $x= \frac{1}{z}$ and $y=f(x)$, show that $$\frac{ \textrm{d}^2f}{\textrm{d}x^2 }= 2z^3 \frac{ \textrm{d}y}{\textrm{d}z} + z^4 \frac{\textrm{d}^2y}{\textrm{d}z^2}$$
1
vote
3answers
32 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)} = ...
2
votes
1answer
19 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
17 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
34 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
26 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
19 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 ...