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

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3
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0answers
16 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
59 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?
-2
votes
0answers
8 views

First and second derivative of a Lagrange polynomial? [on hold]

How can I calculate the first and second derivative of a Lagrange polynomial? Thanks a lot
0
votes
0answers
16 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
32 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
34 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
49 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 ...
2
votes
1answer
58 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 ...
3
votes
1answer
36 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 $\vec{r}$, ...
0
votes
2answers
22 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
43 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 ...
0
votes
2answers
49 views

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

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
30 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)} = ...
-1
votes
0answers
55 views

How to use first derivative of a square root to find $x$ [on hold]

I wanted to differentiate the pythagorean theorem to find the hypotenuse of a triangle, but did not know it's height exact value, which is $(100-x)$. The theorem gave me the equation $$ C=\sqrt{125^2 ...
2
votes
1answer
18 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
16 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
31 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
24 views

derivative of a linear operator

I am a little confused by the proof in the picture. Doesn't the calculation show $dB(u)h = B(h_1,u_2)+ B(u_1, h_2)$?
-1
votes
0answers
29 views

derivative of exponential function proof- [on hold]

I will post link because stackE won't accept size or type of file. This is part of proof for derivative of exponentinal function. 1.part http://postimg.org/image/iz97da581/ How do they get bottom ...
0
votes
0answers
14 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
42 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
23 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
69 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 ...
-5
votes
0answers
28 views

Solving derivative problem [on hold]

Derivative of f(x) = x.cosx/sqrt(1+x^2)?
0
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2answers
79 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 ...
2
votes
0answers
32 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 ...
0
votes
0answers
48 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 ...
0
votes
2answers
52 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
28 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 $H$ ...
1
vote
1answer
35 views

What is the derivative of the ReLu of a Matrix with respect to a matrix

I want to compute $\frac{\partial r(ZZ^tY)}{\partial Z}$ where the ReLu function is a nonlinear operator $r(x)=max(0,x)$ and $Z \in\mathbb{R}^{n\times m}$ is a matrix. I am wondering also if the ...
3
votes
3answers
66 views

Calculate derivative of integral

I tried to calculate the derivative of this integral: $$\int_{2}^{3+\sqrt{r}} (3 + \sqrt{r}-c) \frac{1}{2}\,{\rm d}c $$ First I took the anti-derivative of the integral: ...
4
votes
3answers
53 views

Derivative of $f(t)=\frac {1}{\rho}\log (1+\rho t)$

Could you have me to find the ferivative of $$f(t)=\frac {1}{\rho}\log (1+\rho t)$$ with repsect to $t$? And Is it $$\lim_{t\to \infty} \frac {f'(t)}{t}=1?$$ Update: Based on Hint of Surb: ...
2
votes
1answer
36 views

third derivative of inverse function

Is my way of solving and my answer correct? Let $f(x)=x+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}+\frac{x^5}{5}$ And $g(x)=f^{-1}(x)$ Find $g'''(0)$ My attempt: We know that ...
2
votes
1answer
42 views

Proving that a polilinear operator is differentiable

A Polilinear map operator is $P:X^1 \times ... \times X^n \to Y$ such that the foolowing applies: $\lambda, \mu \in R$ $$ P( \lambda x_1^1 + \mu x_2 ^1, x^2,...,x^n)= \lambda P(x_1^1,x^2,...x^n)+ \mu ...
0
votes
2answers
23 views

continuously differentiable multivariable functions

What does it really mean to say a function $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$ is continuously differentiable? A function $f:\mathbb{R}\rightarrow \mathbb{R}$ is continuously differentiable if ...
1
vote
5answers
69 views

Does the function $|x^2-4|/x$ have critical points?

Does the function $|x^2-4|/x$ have critical points? I tried differentiating and putting the derivative equal to 0.But I'm still a bit confused (as I got no solution).
2
votes
1answer
30 views

Derivative of Bezier Rectangle

From this page Derivatives of a Bézier Curve, I can see that the derivative of a degree $N$ Bezier curve is just a Bezier curve of degree $N-1$ and it explains how to calculate the control points by ...
0
votes
0answers
20 views

Rankine Hugoniot, taking limits

I have seen two different derivations of the Rankine Hugoniot jump conditions across a shock s(t) in the xt-plane. I present a summary of the two different derivations and then post my question in ...
0
votes
1answer
27 views

How to differentiate with respect to component of a vector?

Let $\vec{\alpha}=\frac{m(\vec{x})}{x^2}\vec{x}$ where $\vec{x}=(x_1,\,x_2)$. In a book I read in Eq.(3.24), it was given that $$ \frac{\partial \alpha_1}{\partial x_1}=\frac{d m}{d ...
4
votes
2answers
64 views

$n$-th derivative of $(x^2-1)^n$ has distinct real roots in $[-1,1]$.

For $n=1,2,3,\ldots$, let $$f(x) = (x^2-1)^n .$$ Show that the $n$-th derivative $f^{(n)}$ has distinct real roots in $[-1,1]$. I have no idea about the problem. Could I have a hint?
2
votes
2answers
47 views

Second differential of the norm in an infinite dimensional Hilbert space

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 [solved] So we can ...
0
votes
0answers
16 views

Probability distribution of derivative of function of random variable

The calculation of probability distribution of a function of random variable is a well established theory and there are general rules on how to go from the distribution of r.v. to the distribution to ...
0
votes
2answers
43 views

Maximum value of $f(x)=\frac{x^2}{x^3+200}$ over natural numbers

This was a great problem I came across today.Just wanted to share it :-) $f$ is a function defined over the set of natural numbers(I mean the domain is natural numbers) by ...
1
vote
1answer
37 views

Pathological Question involving $C^1$ Criterion for Differentiability

Edwards1973 gives a sufficient condition for differentiability: If all partial derivatives of $f$ exist at every point of an open set containing $\vec a$, and the partials are continuous at ...
0
votes
1answer
35 views

Derivatives Must Exist over an Entire Open Set?

Let the domain of some function $f$ be an open set that contains the point $a$. This question is with regards to the domain of the (partial) derivatives of $f$. Claim: If all the partial derivatives ...
2
votes
2answers
65 views

How do I differentiate the following implicit function

$$\sqrt{x^2+y^2} = e^{\sin^{-1} \left( \frac{y}{\sqrt{x^2 + y^2}} \right) } \\ \text{find} \quad \frac{d^2x}{dy^2} \quad \text{i.e. prove} \\ \frac{d^2x}{dy^2} = \frac{2 \left( x^2 + y^2 \right) ...
2
votes
1answer
42 views

Number of points of discontinuity

Find the number of points where $$f(\theta)=\int_{-1}^{1}\frac{\sin\theta dx}{1-2x\cos\theta +x^2}$$ is discontinuous where $\theta \in [0,2\pi]$ I am not able to find $f(\theta)$ in terms of ...
0
votes
0answers
11 views

Differentiate function returning vector

can I differentiate a function which is returning a vector? I'm trying to implement Least Squares method on sets of points, but I'm stuck at defining Jacobian, which is numeric, but then, I have no ...
3
votes
4answers
77 views

How to prove that the line perpendicular to the radius is the tangent in the calculus sense?

Let $P=(p_1,p_2)$ be a point on an semicircle and $r$ be the line perpendicular to the radius $\overline{OP}$, like the picture below. Euclid showed (Book III, Proposition 16) that $r$ does not ...
0
votes
0answers
22 views

General formula for sinusoidal taylor series centered at any a?

I understand that to find a taylor series centred at a particular a value you need to find a formula for the nth derivative, but this is tricky for cos(x) and sin(x). Is it possible to have a formula ...