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

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32 views

Differentiability of non-analytic complex functions

Any complex function that is analytic on an open set is differentiable on that set. But can a function fail to be analytic on an open set but still be differentiable? For example, the function ...
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2answers
51 views

Properties inherited by $f\circ g$ from $f$

Suppose $f,g:\mathbb{R}\to \mathbb{R}$ Prop: Suppose $g$ and $f \circ g$ are ______, and $g$ achieves every value in $\mathbb{R}$. Then $f$ is ______. If in the blanks we put the word ...
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1answer
7 views

Sufficient criterion for a function in C to be differentiable

Give a sufficient criterion for a function f(z), z $\epsilon$ C to be differentiable at $z=z_0$. I know that continuity does not imply differentiability, can't think of a criterion that implies ...
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3answers
44 views

Implicit differentiation of trig functions

I'm struggling somewhat to understand how to use implicit differentiation to solve the following equation: $$\cos\cos(x^3y^2) - x \cot y = -2y$$ I figured that the calculation requires the chain ...
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1answer
24 views

Limit involving directional derivatives [closed]

If $z = f(x, y)$ is differentiable at $\textbf{x}_0 = (x_0, y_0)$ Find $ \lim\limits_{\textbf{x} \to \textbf{x}_0}\dfrac{f(\textbf{x})-f(\textbf{x}_0)-\nabla f\left(\textbf{x}_0 \right)\cdot ...
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1answer
24 views

Solve given qus without using partial fraction method

$$z=f\left(x,y\right)=x^{2}\tan^{-1}\left(\frac{y}{x}\right)-y^{2}\tan^{-1}\left(\frac{x}{y}\right)$$ Prove that $$\frac{\partial^{2}f\left(x,y\right)}{\partial x\,\partial ...
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4answers
149 views

derivative of $x\cdot|\sin x|$

I have the function $f(x)=x|\sin x|$, and I need to see in which points the function has derivatives. I tried to solve it by using the definition of limit but it's complicated. It's pretty obvious ...
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3answers
62 views

Interesting, unusual max/min problems?

So I've got to that stage of my elementary mathematics subject for engineers when we talk about differentiation and solution of max/min problems. And I'd like to entertain and engage the students ...
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2answers
50 views

Successive Differentiation of $\mathrm{e}^{g(t)}$

I am trying to find the closed for solution for $A_n$. Assume $A_0 = g'(t)$, $A_1 = g'(t)$, and $$\dfrac{d^n}{dt^n}\left[e^{g(t)}\right] = A_n e^{g(t)}$$ The problem has a recursive relationship of ...
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1answer
20 views

Calculate derivative of multicase function involving exponentials as $x \to 0$ by definition

While this seemed (and probably does seem to some of you) like a simple question a first it stumbled me a bit. We were asked to calculate the derivative of: $$f(x) = \left\{ \begin{array}{lr} ...
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1answer
30 views

Prove there's $M>0$ such that: $f(x)\le Mx^2$

Let $f:[-1,1]\to\mathbb{R}$, three-times differentiable function and $f(0)=0$, $f(x)\ge 0$ for all $x\in[-1,1]$. Prove there's $M>0$ such that $f(x)\le Mx^2$. Hint: use Taylor formula. So ...
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0answers
67 views

A Tricky Weak Derivatives question

I recently came across the following statement and am having trouble proving it correct. I wonder if you could help. Given a weak derivative, $x'$, there exists an absolutely continuous ...
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0answers
25 views

Derivative with respect to a function

We have a function ${f(s,{\psi(s)}_{3\times 1})}_{3\times1}\tag1$ Given Data $f,\psi$ are matrices and their dimensions are already given in the question s is not a matrix, it is a scalar ...
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1answer
19 views

Differentiation under the integral sign (one complex variable)

Let $u(z), u'(z)$ be complex-analytic functions on an open neighborhood $\Omega \subseteq \mathbb{C}$ of the origin. Also, let $f(X)$ be a complex-analytic function. For $s \in [0,1],$ define $$g(s,z) ...
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0answers
40 views

Function with constant derivative

We have a column matrix $P_i$ defined as follows $P_i= {\begin{pmatrix} a_i \\ b_i \\ c_i \end{pmatrix}}_{3\times 1}\tag 1 $. Given Data All $a_i,b_i,c_i$ are constants It is given that $i$ can ...
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1answer
50 views

Finding the derivative of $y=12x^4\sqrt[3]{x^2}-2e^x+9$

Let $$ y=12x^4\sqrt[3]{x^2}-2e^x+9 $$ How can we find $y^\prime$?
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1answer
26 views

How to derive “Pooled Sample Variance”?

Let $s_p^2 = bs_1^2 + (1-b)s_2^2$, this can be an unbiased estimator of population variance, provided we find the correct value for $b$; in particular, $s_p^2 = \frac{(n1-1)s_1^2 + ...
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1answer
20 views

Derivative of an integral with variable in upper bound and a term of the integrand

So I want to take the first and second derivatives of a function g(Z) which is made up of several terms, one of which is where Z and H are our variables. Taking the derivative of this, it seems ...
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0answers
18 views

Derivatives of Lagrange polynomials

It seems there is some relationship between Lagrange polynomial and Legendre polynomial. That is Lagrange polynomial can be expressed as a function of Legendre polynomial. If so, I could use this ...
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2answers
41 views

Calculus - Derivative help.

I need to get dervative for this function $$\sqrt{1+\sqrt{x}}'$$ I used $(f+g)'(x) = f'(x) + g'(x)$ so: $$\sqrt{1} + \sqrt {\sqrt{x}}$$ So : $$\sqrt{1}' = 1' = 0$$ $$\sqrt {\sqrt{x}'} = \sqrt ...
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2answers
100 views

How do i construct $C^\infty$?

I'm trying to define $C^\infty$ rigorously and i have a trouble with this. Mathematical Induction should be used, but i dunno where to apply this. I'm going to illustrate what i tried below: Before I ...
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1answer
51 views

If function $f$ has zero value and positive derivative at both endpoints, then $f''(\eta)=f(\eta)$ for some $\eta$ [duplicate]

Suppose $f(x)$ is differentiable on $[a,b]$, twice differentiable in $(a,b)$. Given that $f(a) = f(b) = 0$, $f'(a)f'(b) > 0$, Prove that $\exists \zeta \in (a,b), f(\zeta) = 0$ and $\exists \eta ...
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1answer
33 views

Differentiate $(x-1)^2 \sin x$ where $x$ is in radians

How would I differentiate, simplify and then find $f'(\pi/2)$: $$ f(x)=(x-1)^2 \sin x $$ I'm not sure how to differentiate $\sin x$ to then use it later to find an answer, any help would be much ...
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2answers
51 views

Second derivative using limits

If f is a function that is two times differentiable at x = a then: $\lim\limits_{h \to 0} \frac{f(a+h)-f(a)-hf'(a)}{h^2/2}=f''(a)$ I don't know how to prove or disprove this. I know I have to use ...
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1answer
22 views

Non-differentiability of $\max \limits_i f(i)$

How can we formally show that $\max$ and $\min$ functions are non-differentiable? In particular, I was looking at the L1 matrix norm defined as: $\|A\|_1 = \max \limits_{i \le j \le n} \sum ...
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3answers
73 views

Find the limit and derivative of integral function.

$\psi_m(x)$ is defined as $$\int_0^{\ln|x|}e^{mt}\sin(t)^m\mathop{dt}$$ with $m$ a natural number greater then zero. Now the question is, does $\lim\limits_{x\to 0}\psi_m(x)$ exist. I've tried using ...
5
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1answer
81 views

Derivative of $\frac{x}{f(x)}\frac{df}{dx}$

Suppose we have a function $f(x):\mathbb R^+\to\mathbb R^+$ that satisfies: 1) $0\leq\frac{df}{dx} \leq 1$ 2) $f(0) = 0$, then do we have $$\frac{d}{dx}\left(\frac{x}{f(x)}\frac{df}{dx} ...
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1answer
34 views

Using bordered Hessian matrix to determine non-degeneracy and type of constrained extremum

I have the following problem: $\def\f{f(x_1,x_2,x_3)}\def\1{x_1}\def\2{x_2}\def\3{x_3}\def\n{\nabla}\def\g{g(x_1,x_2,x_3)}\def\l{\lambda}\def\q{\begin{pmatrix}}\def\p{\end{pmatrix}}$ Find the ...
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1answer
16 views

How do I find the ridges and valleys given a surface elevation function

Given a surface with a single elevation value for every x and y how can I find the places where the isoelevation contours have the tightest bends? And how can I differentiate between bends that are ...
0
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1answer
39 views

A rigorous proof of continuous differentiability

This small step comes from my reading on saddle point approximation: suppose $$ w=\text{sign}(s)\sqrt{2(s K'(s)-K(s))}\tag{*} $$ where $K$ is convex with and continuously differentiable for all orders ...
2
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1answer
60 views

What is an intuitive way to see $\frac{d}{dx}\sin^{-1}x+\frac{d}{dx}\cos^{-1}x=0$?

Without calculation, explain why $\frac{d}{dx}\sin^{-1}x+\frac{d}{dx}\cos^{-1}x=0$?
4
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3answers
62 views

Differentiate with product rule

Question: differentiate $x(x^2 +3x)^3$ I have gotten to the point where i've used the product rule and i've gotten $$(x^2 + 3x)^3 + x\cdot(3x+9)(x^2 + 3x)^2$$ but now that it comes to the ...
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2answers
59 views

Find the Derivative

I'm currently studying the product rule and have come across a section of questions that seems to make no sense. I'm sure there's just one little thing that I'm missing but I am unable to spot it. ...
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2answers
52 views

Definite integral-dot product

I have an integral equation containing dot product $$\int_{0}^{L} \left(\frac{a}{L}.b(s)\right)\mathrm ds\tag 1$$ Data Given a is a constant vector of size 3 b(s) is a varying vector of size 3 " . ...
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4answers
45 views

Proving the Derivative of $f'(x) = b^x$

Given $f(x) = b^x = e^{x\ln b}$ for $b > 0$, can someone show me how $f'(x) = \ln b e^{x\ln b}$ ?
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0answers
25 views

Sufficient conditions for the objective function to have gradient pointing towards the origin

Say I have a sufficiently smooth objective function $J(x)$. How can I ensure the below statement is fulfilled -with additional assumptions to $J$ if needed. $\underset{{{| x ...
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2answers
88 views

How to obtain probability distribution from the generating function $G(s) = e^{a(s-1)^2}$?

I was trying to get the probability distribution $p(n)$ from a generating function $G(s)$ like this: $G(s) = e^{a(s-1)^2}=\sum s^np(n)$ I need first to do Maclaurin expansion of the exponential and ...
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1answer
63 views

Computing the derivative of a transformation matrix

I am trying to find a geometric transformation between two images, where the transformation is a simple scaling matrix. So, if I denote the two image functions as $r$ and $f$ and the scaling matrix as ...
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5answers
182 views

Is there an easier way to find $\frac{\mathrm d^9}{\mathrm dx^9}(x^8\ln x)$ than using the product rule repeatedly?

Find $\dfrac{\mathrm d^9}{\mathrm dx^9}(x^8\ln x)$. I know how to solve this problem by repeatedly using the product rule, but I was wondering if there is a short cut. Thanks.
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0answers
19 views

Applications of Continuity and Differentiability on a Tough Qn

Given f is cont on [0,1] and that it is twice differentiable on (0,1). Suppose that Integral from 0 to 1 of f(x) dx = f(0) = f(1). Prove that there exist a number c where c is an element of (0,1) ...
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3answers
35 views

For what values of $r$, $x^r$ has infinite slope at $x=0$?

I'm learning calculus form MIT OCW 18.01SC. In session 23 (it's about linear approximation), prof computes linear approximation near $0$ of some basic functions. $$\sin{x}, \cos{x}, e^x, \ln{(1+x)}, ...
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1answer
23 views

Baby version of Sturm Comparison Theorem

In Problem 15-32 of Spivak's Calculus, 4th edition, he proves the following: Suppose $\phi_1$ and $\phi_2$ satisfy $$\phi_1''+g_1\phi_1=0, \\ \phi_2''+g_2\phi_2 = 0,\\[10pt] g_2>g_1, \\[10pt] ...
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2answers
34 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 $$V (x, y) = U (x, y)e^{−ax−by}$$ ...
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1answer
42 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 $\frac{dy}{dx} = .75 -.005y$ ...
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1answer
36 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 ...
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2answers
42 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 $\;g'(4) = 2$.
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2answers
59 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 ...
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2answers
35 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 $f''(t)$ ?
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49 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.
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2answers
90 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 ...